01 - Combine warm water, 1 tablespoon 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 lukewarm milk, remaining sugar, egg, softened butter, and vanilla extract in a large mixing bowl until smooth. Pour in the activated yeast mixture and stir to combine.
03 - Gradually incorporate flour and salt into the wet mixture, stirring continuously until a sticky, shaggy dough forms.
04 - Turn dough onto a lightly floured surface and knead for approximately 5 minutes until smooth, elastic, and no longer sticky.
05 - Place dough in a greased bowl, cover tightly with plastic wrap or a damp towel, and let rise in a warm, draft-free area for about 2 hours until doubled in size.
06 - Punch down risen dough and roll out on a floured surface to 1/4-inch thickness. 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.
08 - Fry beignets in batches, turning occasionally, until puffed and golden brown on both sides, approximately 2–3 minutes per batch. Avoid overcrowding the pot.
09 - Remove beignets with a slotted spoon and drain briefly on paper towels. Generously dust with powdered sugar while still warm and serve immediately.