Kambiz Karami at Anthropology.net (a most interesting blog to follow) mentioned this week that Libyan pottery remains, maybe as old as 10,000 years ago, have provided the oldest evidence of vegetable cooking of we know of, at least in pots (see below for the disclaimer). The remains indicate cooking of vegetables and meat and are associated to pictures of people gathering plants, as well as grinding stones (hand mills) with remains of such provisions.
Ref. Nature Plants.
Oddly enough, just two days later, he contradicted himself, mentioning that the charred remains of nuts and seeds from Palestine, dated from 780,000 years ago, actually provide the oldest evidence of vegetable cooking, even if it's clearly a more primitive way of cuisine and not yet at all the "refined" stew of ancient Libyans, which won them three stars in the Paleo-Michelin Guide for Nomads and five pitchforks in Popular Mesolithic Cuisine.
Update (Jan 4): origin of Libyan and West Asian pottery could be Sudan
Jm8 mentions it in the comments, referencing to Anthromadness. I would need more data to judge (the ref. paper is behind paywall) but it does sounds as probably correct to me, being Sudan/Nubia one of the earliest Mesolithic areas in the Western half of the Old World, one that is way too often and unfairly neglected but that definitely influenced the Levant prior to the development of Neolithic proper (what should explain a lot of things both linguistic and genetic).