01 - Combine warm water and yeast in a small bowl. Let stand for 5 minutes until foamy.
02 - In a stand mixer bowl, whisk together flour, sugar, and salt.
03 - Add the yeast mixture, lukewarm milk, eggs, melted butter, and vanilla extract to the flour mixture. Mix with a dough hook on medium speed for 5–7 minutes until a 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, or until doubled in size.
05 - Punch down the dough and turn it onto a floured surface. Roll out to about 1/2 inch thickness. Cut into 2-inch squares using a sharp knife or pastry cutter.
06 - Place the squares on a parchment-lined baking sheet, cover lightly, and let rise another 30 minutes.
07 - In a large, heavy-bottomed pot, heat vegetable oil 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 a slotted spoon and drain on paper towels. While still warm, generously dust beignets with powdered sugar.
10 - Serve immediately for best texture and flavor.