In a medium bowl, combine warm milk, yeast, and 1 tablespoon of sugar. Stir gently and let it sit for 5–10 minutes until foamy.

To the yeast mixture, add the egg, vanilla extract, and melted butter. Stir until combined.

In a large mixing bowl, whisk together the flour, salt, and remaining tablespoon of sugar. Gradually add the wet mixture to the dry, stirring until a shaggy dough forms.

Knead the dough by hand on a floured surface or with a stand mixer fitted with a dough hook for about 5–7 minutes, until smooth and elastic.

Place the dough in a greased bowl, cover with a clean towel or plastic wrap, and let it rise in a warm spot for 1–1.5 hours, or until doubled in size.

Punch down the dough and roll it out on a floured surface to about ½-inch thickness. Cut into 2–3 inch squares using a knife or pastry cutter.

Heat about 2–3 inches of vegetable oil in a heavy-bottomed pot to 350°F (175°C). Fry the beignets in batches, about 2–3 minutes per side, or until golden brown and puffed.

Use a slotted spoon to transfer beignets to a paper towel-lined plate. While still warm, dust generously with powdered sugar.