Ő főz.

Maybe because "egy" is also "one" and it sounds a bit odd to quantify a person, even subconsciously. Also, what I tend to say regarding this question: with "egy", you introduce one new, well-defined "instance" while without "egy", you define stuff by what is it like rather than creating distinct instances. "(s)he has everything by we identify cooks" rather than "(s)he is one (newly mentioned) instance of cooks".