Eggs are not homogeneous substances or mixtures for what the word "melt" even makes sense. But also, most of an egg's mass is composed of substances that suffer chemical transformations in much lower temperatures than other parts can melt (at any pressure).
Even if there is no oxygen on your environment, an egg will decompose itself before all of it is molten.
Interesting. I assumed that if rocks can melt, so would any substance given enough heat. Granted, not without chemical interactions (I assumed burning wouldn't count), but with protein they would at least unfold (denaturate).
This paper seems to at least hint at the possibility:
