Combine warm water, yeast, and 1 teaspoon of granulated sugar in a large bowl and let it sit for 5-10 minutes until foamy to activate the yeast.

Add warmed milk, remaining granulated sugar, egg, flour, salt, and melted butter to the yeast mixture, then mix until a shaggy dough forms.

Knead the dough on a lightly floured surface for 5-7 minutes by hand or 3-5 minutes with a stand mixer until smooth and elastic.

Place the dough in a lightly oiled bowl, turning to coat, then cover and let it rise in a warm place for 1-1.5 hours until doubled in size.

Gently punch down the dough, then roll it out to 1/4-inch thickness on a lightly floured surface and cut into 2-inch squares.

(Optional) Allow the cut beignet squares to rest, covered, for another 15-20 minutes for extra puffiness.

Heat 3-4 inches of vegetable oil in a large pot or Dutch oven to 350°F (175°C), using a candy thermometer to monitor the temperature.

Carefully drop 3-4 beignet squares into the hot oil and fry for 1-2 minutes per side until golden brown and puffed, ensuring not to overcrowd the pot.

Remove fried beignets with a slotted spoon and place them on a wire rack lined with paper towels to drain excess oil, then generously dust with powdered sugar while warm.

Continue frying the remaining dough, adjusting the oil temperature as needed.