Combine warm water and sugar in a small bowl. Sprinkle yeast over the water, stir gently and let it rest for 5-10 minutes.

In a large bowl, whisk together buttermilk, egg, and melted butter. Add the frothy yeast mixture and stir. Gradually mix in flour and salt until a soft dough forms.

Turn the dough onto a floured surface and knead gently for 5-7 minutes. Place in a greased bowl, cover, and let it rise for 1-2 hours.

Punch down the risen dough, roll it out to 1/2-inch thick, and cut into 2-inch squares. Let the squares rest for 15 minutes.

Heat oil to 350°F (175°C) and fry the dough squares for 1-2 minutes on each side until golden brown. Drain on a wire rack.

Whisk together powdered sugar, milk, and vanilla extract for the glaze. Adjust consistency with milk as needed.

Dip warm beignets into the glaze, ensuring an even coating, and let them sit on the wire rack to set before serving.