01 - Combine warm water, 1 tablespoon of sugar, and yeast in a small bowl. Stir gently and let stand for 5–10 minutes until the mixture becomes foamy and bubbly.
02 - Whisk together the milk, remaining sugar, egg, softened butter, and vanilla extract in a large mixing bowl until fully incorporated. Pour in the activated yeast mixture and stir to combine.
03 - Gradually add the flour and salt to the wet mixture, mixing continuously until a sticky, cohesive dough forms. Scrape down the sides of the bowl as needed.
04 - Turn the dough out onto a lightly floured surface and knead for approximately 5 minutes until the dough becomes smooth, elastic, and no longer sticky to the touch.
05 - Place the dough in a lightly greased bowl, cover with plastic wrap or a damp kitchen towel, and let rise in a warm, draft-free area for about 2 hours or until doubled in volume.
06 - Punch down the risen dough to release air bubbles. Roll out on a floured surface to approximately 1/4-inch thickness and cut into 2 1/2-inch squares using a sharp knife or pizza cutter.
07 - Heat vegetable oil in a deep fryer or heavy-bottomed pot to 350°F. Fry the beignet squares in batches, turning occasionally, until puffed and golden brown on both sides, about 2–3 minutes per batch.
08 - Remove beignets with a slotted spoon and drain briefly on paper towels to absorb excess oil. Generously dust with powdered sugar while still warm and serve immediately.