Abbott's Lobster in the Rough, it is primarily to get their steamed lobsters. No matter how much practice we get at home, we never seem to get the lobsters as good as they do. I heartily recommend Abbott's for that, for the awesome bisque (above) and for its wonderful setting on the docks in Noank.
However, I have a bone to pick with them about their hot lobster rolls. Why, why, why do it in the hamburger roll? It looks cool (see above) but is utterly impractical. The meat falls out, the butter doesn't work, etc. It is not a pleasant eating experience. A cold lobster roll might actually work this way, since the mayo binds the pieces together, keeping them in the roll like a chicken salad sandwich. But I wouldn't put a bunch of tiny pieces of chicken sans mayo between two pieces of loose bread, and the same goes for lobster. Get some nice small grinder rolls, toast them, and then you can compete for the best hot lobster roll in Connecticut (see below).