Several of these I just list appetizers/small portions. The best way to eat NOLA is to grab small plates/share and go to the next spot. Only have 1 big meal a day. Any po boy or burger will sit heavy, so keep that in mind and consider sharing.
Johnny's Po-Boys - any are good but I love the alligator sausage
Mahony's Po-Boys - now has a location in the quarter. Debris fries and a hot sausage po-boy please, extra mayo and pickles.
Port of Call - get here 10 minutes before the 11am opening, sit at the bar, grab a mushroom cheeseburger, loaded baked potato, and wash it down w/a monsoon.
Cane and Table - great brunch and cuban influenced drinks.
K-Pauls - jalapeno cheddar rolls, turtle soup, surf-n-turf.
Dragos - charbroiled oysters and crawfish mac-n-cheese
Muriel's - grab a Sazerac at the bar and head upstairs to the seance room, finish drink on the balcony overlooking Jackson Square. At night you'll usually have it to yourself and it is quiet and beautiful.
The Corner Oyster Hopuse off Jackson Square corner of Decatur and St. Peter - never eaten here but excellent bloody mary.
Evangeline - if they have the blackened bologna on the menu order it and a side of blue plate mayo.
Jacques Imos - paneed rabbit and alligator shrimp cheesecake. (not in the quarter)
Bourbon House - charbroiled oysters and crab fingers bordelais, milk punch
French Market Restaurant and Bar - BBQ shrimp and maque choux.
Deanies - BBQ shrimp
Emeril's NOLA - go to the bar, grab a drink, or Ms Hattie's Wings (chicken wing stuffed like an egg roll) and New Olreans Hot Frog Legs (badass - think hot chicken).
Mr. B's Bistro - shrimp and grits.