In a small bowl, combine 1/4 cup of warm water with 1 tablespoon of sugar. Sprinkle active dry yeast over the top and let it sit for 5-10 minutes until frothy.

In a large bowl, whisk together buttermilk, egg, and melted butter until smooth. Gradually add the yeast mixture and a pinch of salt. Slowly incorporate flour until a sticky dough forms.

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

Punch down the dough, roll it out to 1/2-inch thickness, and cut into 2-inch squares.

Heat oil to 350°F in a deep pot. Add dough squares in batches and fry each side for 1-2 minutes until golden brown. Transfer to a wire rack to drain.

In a medium bowl, whisk together powdered sugar, milk, and vanilla extract until smooth. Adjust milk for desired consistency.

Dip warm beignets into the glaze. Allow excess to drip off and place back on the rack to set. Serve warm.