Activate the Yeast: In the bowl of a stand mixer, combine the warm water, yeast, and half of the granulated sugar. Let the mixture sit for about 5 minutes until a thick, foamy layer develops on the surface.

Combine Wet Ingredients: Once the yeast is bubbly, whisk in the milk, vanilla extract, the remaining granulated sugar, and the egg until well combined.

Initial Flour Mix: Add 1 3/4 cups of the all-purpose flour to the wet mixture and whisk until the batter is smooth.

Knead the Dough: Attach the dough hook to your mixer. Add the remaining flour and the salt, mixing on medium speed. Gradually add the softened butter one tablespoon at a time. Continue mixing for about 4 minutes until the dough is smooth and starts to pull away from the sides, though it will remain slightly sticky.

The First Rise: Transfer the dough to a lightly oiled bowl. Cover it and place it in a warm, draft-free spot to rise until it has doubled in size, which usually takes about 2 hours.

Roll and Cut: Turn the risen dough out onto a generously floured surface. Roll it into a rectangle about 1/2-inch thick. Using a pizza cutter, cut the dough into 2-inch to 2 1/2-inch squares.

Fry to Golden Perfection: Heat 3 inches of oil in a large pot until it reaches 360°F on a candy thermometer. Fry the beignets in batches of 3 or 4. Once the bottom is puffed and golden, flip them over to brown the other side.

Drain and Dust: Use a slotted spoon or spider strainer to remove the beignets and place them on paper towels to drain briefly. While still hot, transfer them to a wire rack and dust heavily with powdered sugar. Serve immediately.