Can we revert back a broken egg into the original one? Given that we are allowed to increase entropy in some other part of the system Background (and much of the argument for the question)
The second law of thermodynamics says(as my book states it):

If a process occurs in a closed system, the entropy of the system increases for irreversible processes and remains constant for a reversible process. It never decreases.

Now while looking at some educational videos on YouTube I found that entropy is sometimes related to the amount of disorder in the system. They say this as follows (not the exact statement) :

There are many more disordered states than ordered states and therefore it is much more likely for the entropy to increase or remain the same. Also it is not necessary that entropy cannot decrease, rather entropy in some part of the system do can decrease but the only but only necessity is that increase in entropy in some other part of the system should compensate for it so that there is a net increase in the entropy of the whole system. 

An example of this is that of the formation of crystals where the crystals do become ordered (and hence decrease the entropy) but the heat released in due process compensate for this and increases the total entropy of the system by the following formula:
$$\Delta S = \int_i^f \frac {dQ}{T}$$
Question
It is usually introduced to the concept of entropy by giving the example of breaking of egg. Now the following bothers me:


*

*Is it possible to revert back the broken egg into the original state by whatsoever process that can do it? Given that the net entropy of such a system increases. 

*If possible, then what is that process that can revert back a broken egg into it's original state?
Note that example of video of the egg being rewinded is not in the list of possible answers.
 A: Others have written excellent answers, but I just wanted to make an analogy using a jigsaw.  Imagine if you had a 10,000 piece jigsaw, made from a picture of an egg.  
If that jigsaw was jumbled up, then it is extremely unlikely that any amount of continued jumbling would return it to its completed state.  
However, by adding external energy (in the form of a person who has eaten food), you could sit down and do the jigsaw and return it to it's original state.
The difference between the egg jigsaw and the egg itself is not just that the egg has many many more "pieces" (thinking of all of the molecules of protein from the white and yolk, and fragments of shell) but that the technology exists to join jigsaw pieces together, and the technology does not exist, at this time to reassemble broken shell fragments into a whole shell.  
However, this is purely a practical problem - an engineering problem, you might say.  In theory, just according to physics, reassembling the egg is equally possible as reassembling the egg jigsaw.  In both cases, we reverse entropy by pumping energy into the process, and in both cases it is  extremely unlikely to happen by chance.
A: The other answers to your question should provide you with sufficient insight, summarized as follows: The energy in the system is so partitioned and dissipated following the egg's breakage that the probability of energetic fluctuations perfectly reversing every mechanistic step is practically zero, even though classical dynamics are time-reversible.
To answer your latter question regarding the mechanism of such a "egg un-breaking," it would be the exact same as reversing $t$ in every equation describing the classical dynamics of the system, as classical dynamics are time-symmetric. However, bear in mind that the process is statistically negligible and possibly even mathematically impossible; there is evidence that certain $N$-body problems are irreversible, and that a loss of information occurs as time progresses, though I confess I am not an expert in this field and therefore cannot defend that statement with certainty.
I highly encourage you the ponder the correct interpretation of entropy, as a measure on the [number of] microstates of a macrostate. The reason the entropy of a broken egg is higher than that of an intact egg is because there is only one possible configuration satisfying our definition of an intact egg (namely, that the shell is in one piece), and a innumerably large number of possible configurations satisfying our definition of a broken egg. Note that we don't even need to define the relative energies of either state to make a statement on the qualitative differences in entropy.
The reason the system remains in a higher entropy macrostate is because it is vastly more probable. It is far more likely that the energy fluctuations are small in magnitude and keep the system in the same macrostate, though the microstate changes continuously (for example, the system evolves through different microstates as time progresses to reach the ultimate equilibrium, with the yolk drifting randomly across the floor, but the egg remains in the "broken" macrostate).
A: As a trivial exercise, break the egg, scoop up the contents, use glue to stick the fragments of eggshell together, and insert the contents.  Barring any chemical changes to the contents due to exposure to the atmosphere, you've unbroken the egg, having increased your own entropy considerably.
Now you may argue that the glue joints aren't the same as the original eggshell, so you can instead use chemical processes to reconstitute the chemical bonds. (In principle: I think we'd need some improvements in nanotech to actually do it.)  This involves increasing your own entropy even further.  Likewise any chemical changes in the contents can in principle be reversed, given suitably advanced handwavium.
A: Theoretically, it is possible, at least if by 'original state' you mean 'macroscopically identical' - if you want the microscopic state to be identical, you encounter a problem, that it is impossible to precisely measure the microscopic state, especially after it was altered, so the 'original state' is unknown.
However, practically, we don't have the technological capabilities to merge all the pieces of the eggshell or to fix the organic membranes and separate the mixed contents of the egg.
A: Let us first consider what exactly happens when an egg breaks. Chemical bonds are broken in the egg shell (mainly calcium carbonate) and the energy is converted to heat and sound. The interior of egg once exposed evaporation takes place and some chemical reactions might degrade the yolk.
If we are only concerned about the exterior of the egg going back into the initial form we would need all the energy lost in form of heat and sound to be returned to the shell in exactly the reverse of how it was released. Effectively we want the scenario of what would happen if we played the video of the egg breaking in reverse to actually happen.
Now if we want the interior if the egg also to go back to the initial form then we’d need for the water lost to be gained and the chemical reactions to be reversed. This also has to happen in the exact reverse order.
Those are if you want to go to the exact state as before. However, if you’re satisfied at getting an egg back you can probably conduct “surgery” on the broken egg to bring it back into a functioning egg state.
I find it easier to think about this in terms of a drop of ink in water. Initially, the drop is concentrated around one region. In case of the image, it’s more like a layer of ink. Soon the ink molecules will collide with the water molecules and each other and their velocities completely randomized.

Now if we want to go back to the initial state where the ink and water are separated, then the collisions have to be exactly in the reverse order and direction as to how it got to the current state. Let us say with probability $p$ (consider independent of time for simplicity) a molecule reverses its direction in the time interval $dt$ such that it reaches the state it was at $-dt$. Now for this particle to reverse back to a state at time $-T$, the probability would be $p^{T/dt}$. That was for one particle. Now if you have $N$ particles in your system, that means the probability to reverse state would be given by $p^{N(T/dt)}$. Even if we took $p=0.999$, which in itself is crazy, the total probability would still be ridiculously close to $0$ due to just the number of particles being $N\sim 10^{23}$. Plug it in a calculator and check it out for yourself!
This is the microscopic view of entropy. The disordered/mixed/homogeneous state where things are spread out is way way way more likely (read always) than the initial state where we had separated liquids. However notice that this statement is true for any microscopic state. Even if we began our microscopic state with the ink and water molecules mixed the probability to return to this particular mixed state is zero. The difference being if we look at the macroscopic view a mixture will still look like a mixture. So macroscopically once the mixture appears it’ll keep looking like a mixture even if many collisions are going on in the microscopic world.
As @gardenhead pointed out, this is not the only way for us to get back to the macroscopic initial state of separated ink and water. However, the number of the set of motions that lead to a well separated state are still much less compared to arbitrary motion. Basically what we’re saying is that $p$ is still (much) less than $1$.
A: I assume a chicken egg. A hen can create a new egg that is macroscpically identical to the old one. Elementary particles are indistinguishable so even the fact you're holding the remains of the old egg doesn't matter.
Hens manage to create eggs because they absorb low entropy food and excrete high entropy feces (this way they also manage to keep their bodies in a low entropy state we call "being alive"). This is the other part of the system you're asking about.
So yes, we can revert back a broken egg into the original one. It's lot easier if we're hens.
A: To make a broken egg in reality (so not in a video) return to the non-broken state you have to reverse all the momenta of all the particles that are part of the broken egg, and you have to include all the surrounding particles which are affected by the breaking as well. Including the motions of all particles that constitute you if you are looking at the egg when it breaks. While the egg breaks also photons are emitted. These radiate away at the speed of light, so we can't get a hold of these (or the ones absorbed by matter in the surroundings of the broken egg, e.g. the surface it breaks on). Neither can you use photons that are produced by some source since this alters the surrounding of the broken egg. 
Some kind of future surgery is pure fiction. The broken egg is part of a continuous process in spacetime, including the breaking itself, and you can't isolate the broken egg from that process (governed by the second law of quantum statistical mechanics at the microlevel and the laws of classical chemistry or classical mechanics at the macrolevel). You need to reverse this continuous process, which is an impossibility. 
I can see no means to accomplish this, without changing the broken egg itself, so this will be impossible (and we haven't even taken into account quantum mechanics).
What you ask is somewhat like asking if we can make an egg without the aid of a chicken (even if you let a chicken eat the broken egg the newly created egg isn't the same egg as the broken egg was before it broke). This is obviously impossible like it is impossible to create a living baby without a woman (who has a womb) and man. 
Or take the "simpler" question if you can reverse a flash of lightning. In that case, you have to reverse the increase in entropy (for the whole universe) into a decrease in entropy. This is the main point here. Irreversible processes are...well.. irreversible.
A: If we allow an external environment to interact with our "egg" system, putting our egg back to its original state is the same thing as creating an egg which is identical to the original. The crux of the question is to clarify what we mean by "identical".
In the context of thermodynamics, we cannot possibly mean microscopically identical: if we are to talk about thermodynamic entropy, we must be doing some `coarse-graining', meaning that we identify a single macroscopic state with many ($N$, let's say) sufficiently similar microscopic states. The entropy can then be defined by counting how many many microstates give the same macrostate, $S=\log N$. A state has high entropy if there are lots of microscopic arrangements that look the same macroscopically.
One answer to the question is to start an egg farm, and spend your time checking all the eggs to see if they're the same as the original. Eventually, if you don't run out of resources and patience, you'll find an egg that's sufficiently close to the original that you can't tell the difference. At that point, you have in effect restored the egg to its original state.
