01 - Combine warm water and yeast in a small bowl. Let stand for 5 minutes until foamy.
02 - Whisk together flour, sugar, and salt in a stand mixer bowl.
03 - Add yeast mixture, lukewarm milk, eggs, melted butter, and vanilla extract to the flour mixture. Mix with dough hook on medium speed for 5–7 minutes until smooth, slightly sticky dough forms.
04 - Transfer dough to a lightly oiled bowl, cover with plastic wrap, and let rise in a warm place for 1–1.5 hours until doubled in size.
05 - Punch down dough and turn onto floured surface. Roll out to 1/2 inch thickness. Cut into 2-inch squares using sharp knife or pastry cutter.
06 - Place squares on parchment-lined baking sheet, cover lightly, and let rise another 30 minutes.
07 - Heat vegetable oil in large heavy-bottomed pot to 350°F.
08 - Fry beignets in batches for 1–2 minutes per side until golden brown and puffed. Do not overcrowd the pot.
09 - Remove with slotted spoon and drain on paper towels. Generously dust with powdered sugar while still warm. Serve immediately.