Mrs. BD and I have liked Barbetta for many years. It's on the theater district's Restaurant Row, and it's been in the same townhouse, and owned by the same family, since 1906.
Food is mainly Piedmontese, no red sauces and only one pasta on the menu. I had the rabbit with a white wine sauce as secondi and so did the pupette, while she explained to us the basic architecture of successful playwriting from Aristotle to Beckett (it's always been the same because it works, even for screen writing - we had just seen a fairly OK production of Measure For Measure at The New Vic. She explained that Godot had perfect structure but no content - which was the point. For me, Godot sticks in my head but I don't really want or need it to).
Their pre- or post-theater prix-fixe menu is very reasonable if you pass on their amazing wine list. The upstairs dining is cozy, the downstairs is elegant but simple. The jewels that can be hidden inside simple old brownstones are always surprising to me. They also have a small garden.
Reservations absolutely required and appropriate adult attire is expected but the family which owns it has run a relaxed, highly-attentive, and cheerful, comfortable joint for over 100 years.