I'm bad with restaurants in Manhattan, but Anella and Bozu are two favorites in Brooklyn.
I agree there are many loud restaurants in Manhattan, but that is mostly a factor of the bar and/or a large space (e.g., Buddakan). Look for places not trying to attract attention (a full curtain on the window, rather than trying to advertise their popularity / bar scene).
My main point was probably that Manhattan != NYC. :)