What good is food with no dessert? And do you really need to eat a meal to earn dessert?

In restaurants

Favorite desserts served in restaurants are:

  • Bread pudding. The best type is cooked firm and is usually best served with a sauce. Sometimes the sauce is only buttery, but the best also contains liquor. The best places for this are the Arnaud's,  The Bon Ton, and Pascale Manale's. Commander's Palace has a respectable white chocoloate bread pudding.
  • Banana's Foster. Invented at Brennan's Restaurant in the French Quarter: Banana's are sliced in half and sauteed in a mixture of butter, brown sugar and cinnamon. The finished mixture is then flamed with 151 rum and spooned over vanilla ice cream. The preparation is usually at tableside and is something of a production, but it's well worth it.

Desserts only 

A local favorite is Angelo Brocato's 214 N Carrollton Ave, New Orleans, (504) 486-1465.
The closest you can come to an early 20th century Italian ice cream shop.
It's been in business and gotten the jump on the gelato craze by about one hundred years.
You can get all kinds of Italian ice creams, gelatos, pastries and coffees
Spumoni is a multcolored ice cream that must be tried. Tortoni is a frozen whipped cream concoction which is also popular. If the parking is tight, park in the grocery parking lot a half block away. They also sell a variety of cookies, the fig and seed cookies are very popular.

Coffee and Beignets

Morning Call Coffee Stand is not in New Orleans proper, but in a suburb. It used to be located in the French Market near Jackson Square.
The current location is 3325 Severn Avenue, Metairie Lousiana. 
Here you can get get only cafe au lait and beignets, along with cold milk. The cafe au lait is extremely strong coffee cut 50/50 with milk. Don't ask for it black; it is undrinkable and you will only draw laughter.
The beignets are deep fried dough which puffs up and gets crispy. Ideally, they are served fresh out of the frier and you top them with powdered sugar. You may hav to wait for them to cool to an edible temperature.
Get fresh ones if you can; a good waiter will suggest this. Tip generously. The prices are pretty low. Open 24 hours.

Cafe Du Monde was a competitor to the Morning Call but they are no longer near each other. The main location is directly across decatur street from Jackson Square. I think the Morning Call is a little better, but they are close.
There are also locations in local shopping centers.