Can a microphone on a supersonic aircraft hear noises coming from outside? This webpage includes the following question/answer that includes the italicized text saying that pilots of a supersonic jet cannot hear sounds from outside the aircraft.

Do pilots of a supersonic jet hear the sound of the engines when it
breaks the sound barrier?
Short answer: Yes, pilots of a supersonic
jet can still hear the humming of the engines when their plane breaks
the sound barrier if the sound is transmitted through the air inside
the plane (however, they cannot hear the sounds coming from the
outside). Similarly, the passengers can also hear sounds inside the
cabin, as the air around them (as well as the pilots) is stationary
with respect to the plane.

I comprehend that sounds originating from behind the aircraft cannot catch up to the jet.  However, I think that sounds originating from in front could be heard with a large doppler shift causing the sound frequency to increase.
Am I correct?  Or does the shock wave in front of the jet somehow totally destroy the sound waves?
 A: 
Do pilots of a supersonic jet hear the sound of the engines when it breaks the sound barrier?

Yes, but not through the air.  The sound is largely generated internally to the fuselage of the plane.  While sitting on the deck ready for takeoff, then the engine would likely be much louder but the cockpits are often surrounded by heavy materials reducing sound transmission anyways.  It's just that fighter jet engines are so obnoxiously loud, e.g., see notes about this at: https://physics.stackexchange.com/a/281767/59023.

However, I think that sounds originating from in front could be heard with a large doppler shift causing the sound frequency to increase. Am I correct? Or does the shock wave in front of the jet somehow totally destroy the sound waves?

Your second questions is likely closer to the truth than the former.  The reason being that shock waves cause an abrupt change in the index of refraction for light and for compressive sound waves in a medium.  So much so that sound waves can actually reflect off of the shock wave (e.g., sometimes shock waves reflect off of each other).  Most linear sound waves that start outside of and in front of a supersonic jet would likely be reflected and destroyed.  That is, immediately after reflection the wave would be overtaken by the shock since the sound wave propagates only at the speed of sound whereas the shock speed is defined by the piston/driver (i.e., supersonic jet here) speed.
Regardless of this, one would need a rather intense sound to actually be heard.  That is, suppose we ignore the issues with crossing the shock barrier.  The sound would need a greater intensity than that of the jet, which is not insignificant as I noted above.  Further, the sound would need to be extremely loud to be heard inside the pilots helmet which is intentionally designed to reduce outside noises for communication purposes.  My guess is that something like a missle explosion may be heard, but unfortunately it may have to be close enough to cause damage to the craft.
I will pester a colleague of mine that used to be an air force pilot to get a more definitive answer.
Update
I spoke with my air force buddy and he said a few things that kind of confirmed what I had originally thought.  So most of the noise in the cockpit is due to the engines and it is primarily transmitted through the solid body of the fuselage.  The cockpit is intentionally designed to reduce noise from outside sources.  Further, the helmet has additional noise reduction methods implemented to allow the pilot to mostly hear comms and not much else (besides the vibrations of the engine, of course).
In general, shock waves reflect off of each other so explosions ahead of a supersonic jet might be felt but likely not heard unless they are extreme (in which case the integrity of the jet may be more of a concern than whether the pilot can hear the external noise source).
A: " does the shock wave in front of the jet somehow totally destroy the sound waves?"
I think you might be confused here. The shock waves are sound waves. Remember, sound waves are "air pressure waves", and so are these shock waves. (Ignoring the sound waves inside the metal of the hull - the speed of sound in metal is much higher).
The shock wave effect of a supersonic plane happens because it continuously is generating sound waves, but all these waves arrive at the same time at the stationary observer.
As for the "sounds from in front" - that's rather tricky. The airflow noise is caused by the interaction between the plane and the air, which have quite a difference in speed. This causes a turbulent, noisy boundary layer which interacts with the metal hull - without that hull there would be no boundary layer. This is very much not a localized source.
