Vyaturna Melnitsa — "the Windmill" — is housed in a converted windmill at the entrance to the Old Town. The menu is broader than Neptun's: meze, grills, Bulgarian classics, decent fish, and a stronger wine list with Black Sea regional reds and whites worth ordering by the bottle. The building itself is a converted 19th-century mill, with the original mechanism visible from the dining room.
A good first-night dinner if you have just arrived and want a sit-down meal with a sense of place. Reservations help in August.