Does existence have an essence?

This post assumes distinctions drawn here.

It's sometimes objected that since Thomists say that “that something is” (existence) is really distinct from “what something is” (essence), they can't account for “what existence is”.

The reply is that even though a's existence is distinct from a's essence (a), it doesn't follow that a's existence doesn't have its own essence. So assuming Thomists can give some account of existence (e.g. Barry Miller argues that existence is a property), they can account for “what existence is”.