[nyanmatt]

I believe it’s theoretically possible (practically impossible) if done in vacuum (to prevent oxidation) and you somehow increase the density of the egg to remove any vapor pressure (to prevent vaporization).

Just guessing here. I’m no thermodynamics expert, just a guy who melt things occasionally.

[kergonath]

At atmospheric pressures it would sublimate rather than melt, so it’d be even worse in vacuum. You’d need a reducing atmosphere with high pressures. Carbon is not that easy to oxidise, but there is some oxygen in the egg. So you should be able to do it in a container filled with argon, with an oxygen getter. You’d need to bring it to 106 atmospheres (easy) and 4600 K (really quite hard). The good thing is that at these pressures you should not have issues with the water expanding too much. It’d be better to have a hole in the shell, though.

That is ignoring any interaction with the shell, which should melt around 1600 K.