In a large bowl, whisk together the warm water, yeast, and 1 teaspoon of granulated sugar; let it sit for 5-10 minutes until frothy to activate the yeast.

Add the remaining granulated sugar, milk, egg, salt, and melted butter to the yeast mixture, mixing well. Gradually add flour to form a soft, slightly sticky dough, then knead for 5-7 minutes. Place the dough in a lightly oiled bowl, cover, and let it rise in a warm place for 1.5 to 2 hours until doubled.

Gently punch down the dough, roll it onto a floured surface to about 1/4 inch thick, and cut into 2-inch squares using a pizza cutter or knife.

Heat about 3-4 inches of vegetable oil in a pot to 350°F (175°C), then carefully fry a few beignet squares for 1-2 minutes per side until golden brown and puffed. Remove the beignets with a slotted spoon to drain on paper towels, then generously dust with powdered sugar while warm and serve immediately.