Mahar Bagan
Khayee Road, Khan Laung Quarter, New Bagan
One of the better restaurants in Bagan, with a cheerful and friendly owner who speaks good English and seems happy to indulge customers in stories about the area. The menu consists primarily of Chinese-style dishes. The restaurant serves an excellent array of traditional Burmese food, but you have to visit 4-5 hours in advance to let them know your order, as most Burmese dishes take a long time to prepare.
The Moon Vegetarian Restaurant
North of Ananda temple, Old Bagan
061-60481
Not only the best veggie restaurant in Bagan, but overall just a great eating experience. All food on the extensive menu is freshly prepared, and there's always a special dish of the day. Place sits around ~15pax, so you won't have to sit with the big tourist groups. Costs around 5 USD/pax.