Use a handheld whisk to mix warm water, sugar, and yeast in the bowl of a stand mixer and let it sit for 7 to 10 minutes until a foamy layer forms on top.
Add the milk, butter, egg, vanilla, and salt to the yeast mixture and whisk until well combined. Add the flour to the bowl and gently mix with a wooden spoon until a shaggy dough forms. Some small dry pockets of flour remaining is fine.
Knead the dough with the bread hook attachment on your stand mixer until it pulls away from the sides of the bowl and climbs up the bread hook. Shape the dough into a smooth ball, lightly oil the stand mixer bowl, and place the dough back into the bowl.
Gently move the dough ball around the bowl to coat it in oil. Tightly cover the bowl with plastic wrap and refrigerate the dough for at least two hours or overnight.
Add enough oil to a Dutch oven or large pot to go about 4 inches up the sides. Heat the oil to 360°F, using a candy thermometer to check the temperature.
While the oil heats, roll the dough out on a lightly floured surface into a ¼" thick rectangle. Use a pizza cutter to cut the dough into 2-inch squares. Do not attempt to re-work the dough scraps as it will become tough. Just fry the scrap pieces as they are.
Drop 4 beignets in the hot oil at a time. They will rise to the surface in a few seconds. Cook the beignets for about one minute, flipping them once or twice with tongs after they have puffed up so they brown evenly on both sides. Using a slotted spoon or spider, transfer the beignets to a paper towel-lined baking sheet with a wire cooling rack on top to drain.
Dust the beignets in lots of powdered sugar and enjoy warm with a cup of coffee!