Pour warm (105-110 degrees F) water into a large bowl. Sprinkle on the yeast and sugar and then whisk together. Cover with a light towel or wrap – you do not need to seal it, just cover it to protect from drafts. Let sit for 5-10 minutes. It should become slightly foamy. If it does not foam, than the yeast may be bad, or the water was too hot and killed the yeast.
Add in eggs, milk and vanilla and whisk together. Mix in ½ 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, one tablespoon at a time. Place the dough into a lightly oiled bowl.
Cover the dough with plastic wrap or a warm damp towel, and leave it undisturbed in a warm, draft free place. Make sure you don’t sit it near a fan or an AC vent. Allow the dough to rise until doubled in size (1-2 hours).
Roll out dough to ¼ inch thick. Cut into 2 ½ inch squares. A pizza cutter works well for this.
Heat oil in a large pot to 370 degrees. You should have at least 2 inches of oil for the deep frying.
Deep fry the beignets in batches, don’t crowd. They should pop up when you put them in the oil or 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. It will only take 1-2 minutes per side.
Drain on a plate lined with paper towels. Sprinkle generously with confectioners’ sugar and serve hot.