My impression is that you could hardly find three more different restaurants. Allegretti is very good, serious, Provencal French in a quite formal Flatiron dining room, entrees $30 and up. The chef is a Ducasse, Le Cirque, Atelier alumnus.
Double Crown is loud, raucous, scene-y and serves hit-and-miss Asian fusion dishes and poor cocktails.
Joe Doe is a tiny, casual new American spot, with entrees in the low twenties. I've eaten at the two above but not here.
The chef at Allegretti may be a Ducasse alumnus but ducasse and his organization have no affliliation with the restaurant.