Can I cook an egg by holding it in thermosphere? It is known that in thermosphere, air temperature is rising sharply with increase of altitude.
In upper atmosphere, temperatures can even reach 2000°C or higher: see 2.
However, air there is extremely thin, so heat transfer is very slow.
If I put an egg to the thermosphere and hold it there for a long time at zero airspeed, will it cook?

UPDATE: My own line of thought is that the egg may radiate the heat energy faster that acquiring it. But then why doesn't air around it does not radiate it just as well? Answer: Because of the Sun constantly heating it up. But then the egg should be constantly heated up by the Sun as well as the air around it.
 A: An egg has to reach an inner temperature of 100C to cook and in water the egg shell is kept at 100C for five minutes in a heat bath.
Thus your question is answered by "can the egg shell be heated to 100C by the much hotter thin gas in the thermosphere"
In vacuum the egg will radiate away and go close to 0 kelvin, so in the thermosphere it will be a fight between outgoing radiation to incoming from the "heat bath". Well, the thermosphere does not provide a heat bath.

Although the thermosphere is considered part of Earth's atmosphere, the air density is so low in this layer that most of the thermosphere is what we normally think of as outer space. 

So the answer is that the water from the egg will sublimate through its pores and the rest will be a solid at close to 0K .  
A: EDIT : The egg I am considering doesn't have a shell
Temperature is a measure of the average kinetic energy of particles in a gas. In the thermosphere, the highly energetic solar radiation collide with the (having very less pressure) air, and are thus given very high velocities and so have a high temperature. However, the temperature shouldn't play a big role, especially since the air's pressure is so less.
Since the air pressure is lower, the boiling point of water decreases. Therefore the water inside the egg will just boil out in a short time before the high temperature outside can cook it. So what will remain is the remains of an egg without water and the egg would be at the same temperature as before. The conventional cooking of an egg is more complex, with heat coagulating up the proteins (and therefore the taste ?), the water getting heated up side by side all along gradually. Here the water gets boiled up instantaneously without changing the temperature, and what is left are the remains. The protein denaturization process here will be totally different and so even if the egg manages to get heated up to a high temperature (addressed below), it won't be cooked as you would have expected.
Secondly, the heating process will be totally different. In conventional cooking there are a number of molecules hitting the egg with small velocities while here there will be less number of molecules with extremely large velocities. Therefore the egg will not get uniformly boiled up, but get sparsely collided with molecules of air as well as from solar radiation, whose action might cause the egg to burn up in spots rather than getting cooked up. In short, there is nothing simpler than to put you egg in a cooker. Tossing it up to get fried is not a feasible option in my opinion. :P
