In the bowl of a stand mixer fitted with the dough hook attachment, stir together the granulated sugar, yeast, and ¾ cup of warm water. Let rest for 5 to 10 minutes or until foamy.

Once the yeast mixture is foamy, add the flour, milk, butter, egg, vanilla, and the 1/8 teaspoon of salt. Mix on medium speed until the mixture is tacky and pulls around the dough hook. Remove the dough from the mixer, cover, and let rest in a warm area for up to 2 hours, or until it has doubled in size.

Once the dough has doubled in size, fill a deep fryer, rondeau, or wide, shallow pot with the peanut or canola oil and heat to 350F.

Remove the dough from the bowl and lightly knead it on a floured work surface. Use a rolling pin to roll out the dough until about ¼ inch thick. Using a pizza cutter, cut the beignets into 2- to 3-inch squares. These do not need to be uniform; they are meant to be misshapen.

Working in batches, carefully drop the beignets in the hot oil and fry, flipping once, until lightly golden, about 2 minutes on each side. Repeat to fry the remaining beignet. Put the beignets in a paper bag with the remaining ½ teaspoon salt and the confectioners’ sugar and shake to coat. Serve immediately.