An all time favourite - Riccardo's, 126 Fulham Road SW3. It's a kids/ animals -friendly place so they do get noisy at times and as they are usually well booked, do it a few days in advance.
It's Italian. Lots of really tasty vegetarian things on the menu and a few vegan dishes, and what's vegetarian is easily turned into vegan if you ask. They also specialise in wheat-free and gluten-free pasta.
The portions are rather small though