You can melt anything. An egg will burn first. Then you will get some type of rendered carbon ash. Which will, eventually, melt and/or vaporize with enough heat.

Which raises an interesting question: what if you cooked it in a zero oxygen environment (say argon, nitrogen, or carbon dioxide… basically welding gases because they’re mostly inert). I can’t burn in that context, so does it melt? Or do you drive off all the volatiles and are just left with carbon anyway?