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