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 to flour mixture. Mix with dough hook on medium speed for 5–7 minutes until smooth and slightly sticky.
04 - Transfer dough to lightly oiled bowl, cover with plastic wrap, and rise in warm place for 1–1.5 hours until doubled.
05 - Punch down dough and turn onto floured surface. Roll to 1/2 inch thickness and cut into 2-inch squares.
06 - Place squares on parchment-lined baking sheet, cover lightly, and rise 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 pot.
09 - Remove with slotted spoon and drain on paper towels. Generously dust with powdered sugar while warm.