01 - Combine warm milk, yeast, and 1 tablespoon sugar in a small bowl. Allow to sit undisturbed until foamy, approximately 5 minutes.
02 - In a large mixing bowl, whisk together remaining sugar, eggs, melted butter, salt, and vanilla extract. Stir in the activated yeast mixture.
03 - Gradually incorporate flour into wet ingredients, stirring until a sticky dough forms. Transfer to a lightly floured surface and knead for 5 to 7 minutes until the dough achieves a smooth, elastic texture.
04 - Place dough in a greased bowl and cover tightly. Allow to rise in a warm location until doubled in volume, approximately 1 to 1.5 hours.
05 - Punch down the risen dough. On a floured surface, roll out to 1/2-inch thickness. Cut into 2-inch squares using a sharp knife or pastry cutter.
06 - Heat vegetable oil in a large, heavy-bottomed pot or deep fryer to 350°F.
07 - Working in batches to avoid overcrowding, carefully place beignet squares into hot oil. Fry for 1 to 2 minutes per side, turning once, until puffed and golden brown.
08 - Remove fried beignets with a slotted spoon and transfer to paper towels to drain excess oil.
09 - Generously dust warm beignets with powdered sugar immediately before serving for optimal texture and presentation.