I got a request today regarding a place to eat in New Orleans to impress a "high-falutin'" lawyer from out of town.
I think that in terms of both the quality of the food (and service) and the formality of the restaurant, you can't go wrong with Restaurant August.
Stella is another extremely fancy place to eat. The food is decadent and they are currently featuring a caviar menu. I haven't been here in a couple of years, but I recall that the service was very good and the ambiance was lovely.
If you want to adhere to a classic New Orleans restaurant, I'd suggest Commander's Palace. It's a classic for a reason.