Cochon and Cochon Butcher are really good, non-traditional NOLA spots. Cochon more sit-down, Butcher is a counter-service bar type spot. Pork belly sandwich at Butcher is phenomenal. Located right next to each other in the Warehouse District.
Mr. B's in the Quarter, barbecue shrimp is awesome.
The best breakfast spot in all of NOLA is Russell's Marina Grill up on Lake Ponchartrain. It's about a 15 minute Uber ride from downtown, but their Eggs Ponchartrain, Chicken and Waffles, Bloody Marys, and mimosas with fresh squeezed OJ are well worth the short trip.