A good New Orleans beignet recipe is French Quarter Beignets from the Food Network. Other good recipes include Southern Living's New Orleans Beignets and Epicurious' Buttermilk Beignets.
To make French Quarter Beignets, combine 1 1/2 cups of lukewarm water, 1/2 cup of granulated sugar and 1 envelope of active dry yeast. As the yeast mixture sits for 10 minutes, cream together 2 eggs, 1 cup of evaporated milk and 1 1/4 teaspoons of salt in another bowl. Once the yeast starts to bubble, mix in the egg mixture, 7 cups of bread flour a few cups at a time, and 1/4 cup of shortening or butter. When the dough starts to come together, knead it on a floured surface, and then place it in a covered bowl in a warm location.
After two hours, roll out the dough, and cut it into 1-inch squares that are 1/4-inch thick. Cook the dough in oil at high heat until the beignets are golden brown. After the beignets drain for a few minutes, place them in a bag with 3 cups of confectioner's sugar, and shake the bag until the beignets are thoroughly covered.
Another recipe suggests heating 3/4 cup of whole milk in a saucepan until it is frothy, adding 1 1/2 cups of buttermilk, and then whisking in 4 teaspoons of yeast and 2 1/2 tablespoons of sugar. After five minutes, mix in 3 1/2 cups of bread flour, 1/2 teaspoon of baking soda and 1/4 teaspoon of salt.