Step 1: In a large bowl, combine warm water, sugar, and yeast. Let it stand for about 5 minutes until foamy.

Step 2: In a separate bowl, beat eggs, then add milk, melted butter, and vanilla extract. Mix well.

Step 3: Stir the egg mixture into the yeast mixture, then add salt and 2 cups of flour. Beat until smooth.

Step 4: Gradually add the remaining flour, 1/2 cup at a time, mixing until a soft dough forms.

Step 5: Turn the dough onto a floured surface and knead for about 5 minutes until elastic.

Step 6: Place the dough in a greased bowl, cover with a cloth, and let it rise in a warm place until doubled in size, about 1 hour.

Step 7: Heat oil in a deep fryer or large pot to 350°F.

Step 8: Roll out dough to 1/4 inch thickness on a floured surface and cut into 2-inch squares.

Step 9: Fry beignets in batches, turning them over when they are golden brown, about 2-3 minutes per side.

Step 10: Remove with a slotted spoon and drain on paper towels. Dust generously with powdered sugar before serving.