Pour the warm water into a medium-sized bowl, double check the temperature with a kitchen thermometer to make sure it's not too hot or too cold. Whisk in the yeast and sugar and allow it to sit for 5 to 10 minutes. It should become slightly foamy - or your yeast may be bad.
¾ cup warm water, 2¼ teaspoons active dry yeast, ⅓ cup granulated sugar
Whisk in the eggs, milk, and vanilla.
2 large eggs, ½ cup whole milk, ½ teaspoon vanilla extract
Mix in 2 cups of the flour, then the melted butter, then the rest of the flour. Knead the mixture with your hands until the dough is smooth. If it is too sticky, you may want to add more flour, a tablespoon at a time, until smooth.
4 cups all-purpose flour, 3 tablespoons salted butter
Shape the dough into a ball and place it in a lightly oiled bowl. Cover the dough with plastic wrap or a damp towel and leave it undisturbed in a warm place. Allow the dough to rise until doubled in size, about 1 to 2 hours.
Roll out dough to ¼ inch thick. Cut into 2 ½ inch squares. You'll likely have scraps from cutting the dough, don't try to remix these scraps and cut them into squares, this would result in tough beignets. Fry these scrap pieces just the way they are, they may not need to cook for as long but will be just as delicious!
In a large heavy-bottomed pot, heat oil to 370°F. You want to have at least 2 inches of oil for deep frying.
4-8 cups vegetable oil
Deep fry the beignets in batches so you don’t crowd the pot or cool down the oil too much. They should pop up when you put them in the oil, if they don't, this means the oil is not hot enough. Spoon the oil over the tops of the beignets as the bottoms are frying. Make sure to turn them over so they are golden (not brown) on both sides, about 1 to 2 minutes per side.
Drain on a plate lined with paper towels.
Sprinkle generously with powdered sugar and serve hot.
2 cups powdered sugar