01 - Combine warm milk, sugar, and yeast in a small bowl. Let stand for 5-10 minutes until mixture becomes foamy, indicating yeast activation.
02 - Whisk flour and salt together in a large mixing bowl. Add the yeast mixture, egg, melted butter, and vanilla extract. Mix until a sticky, cohesive dough forms.
03 - Turn dough onto a lightly floured surface and knead for approximately 5 minutes until smooth and elastic, adding minimal flour only if necessary to prevent sticking.
04 - Place dough in a greased bowl, cover with plastic wrap or a clean towel, and let rise in a warm, draft-free area for 1 to 1.5 hours until doubled in volume.
05 - Turn risen dough onto a floured surface and roll to 1/2-inch thickness. Cut into 2 1/2-inch squares using a sharp knife or pizza cutter.
06 - Pour vegetable oil into a deep pot or fryer and heat to 350°F. Use a kitchen thermometer to ensure accurate temperature.
07 - Fry 3-4 beignets at a time for 2-3 minutes per side, turning once, until puffed, golden brown, and cooked through. Maintain oil temperature between batches.
08 - Remove fried beignets with a slotted spoon and transfer to paper towels to drain excess oil briefly.
09 - Generously dust warm beignets with powdered sugar, coating all sides thoroughly. Serve immediately while still warm for best texture and flavor.