Yes, depending on the context this could be taken to mean "he cooks her" too; though I don't think this makes any sense to you yet as you'll learn that form of "la" in the clitics lesson :)
Does the English word really have that meaning? Never heard of it and Collins doesn't know either, is it some slang? In Italian the cooking process is "la cottura" and the cooking time "il tempo di cottura".