In a large mixing bowl or the bowl of a stand mixer, combine the warm water, yeast, and half of the granulated sugar. Set aside for 5 minutes until the yeast becomes foamy.

In a separate bowl, mix together the flour and salt, and set aside.

Once the yeast mixture is bubbly, stir in the milk, vanilla extract, remaining sugar, and the egg. Whisk until the mixture is smooth.

Add half of the flour mixture to the wet ingredients and whisk until well combined and smooth.

Using a dough hook attachment, gradually add the remaining flour while mixing on medium speed. Once the dough begins to form, incorporate the softened butter. Continue kneading for about 4 minutes until the dough is smooth and pulls away from the sides of the bowl. If the dough is too sticky, mix in an additional 1/4 cup of flour.

Transfer the dough to a lightly oiled bowl, cover it, and allow it to rise in a warm place for at least 2 hours. Alternatively, you can refrigerate the dough for up to 48 hours.

Heat about 3 inches of vegetable or peanut oil in a large pot over medium-high heat, using a candy thermometer to monitor the temperature. The oil should reach 360°F.

Roll the risen dough out on a well-floured surface to a 1/2-inch thickness. Cut the dough into 2 to 2 1/2 inch squares using a pizza cutter.

Fry 3-4 beignets at a time in the hot oil. Cook until one side is golden and puffed, then flip them over to fry the other side. Remove with a slotted spoon or spider tool once both sides are golden.

Place the fried beignets on paper towels to drain any excess oil, then dust generously with powdered sugar before serving.