[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.