Like many delicious treats, this preparation takes a bit of time and planning. You can speed up the process of proofing the dough if you leave the dough covered at room temperature for an hour or so, instead of letting it rest in the refrigerator overnight.
Adapted from My New Orleans: The Cookbook by John Besh.
Pour the warm milk into a large bowl. Mix 1 tablespoon of the granulated sugar, the yeast, and a heaping tablespoon of the flour into the milk, mixing with a whisk, until both the sugar and the yeast have dissolved.
Once bubbles have developed on the surface of the milk and it begins to foam, whisk in the butter, salt, and vanilla. Add the remaining flour and sugar, folding them into the wet ingredients with a large rubber spatula. Knead the dough by hand in the bowl for about 5 minutes, then cover the bowl with plastic wrap and refrigerate the dough for 6–8 hours.
Remove the dough from the refrigerator and roll out on a floured surface to a thickness of ¼ inch. Cut into 2-inch squares, cover loosely with plastic wrap, and allow the beignets rise for about an hour.
Heat the oil in a large deep skillet over high heat until it reaches 350 degrees. Use a candy thermometer to check temperature. Fry the beignets in small batches in the hot oil, turning them every 30 seconds or so with tongs, until golden brown all over. Use tongs to remove beignets from the oil and drain on paper towels. Put the powdered sugar into a fine-mesh strainer and dust the warm beignets generously with the sugar.