01 - Combine warm water and yeast in a small bowl. Let stand for 5 minutes until foamy and activated.
02 - Whisk together flour, sugar, and salt in the bowl of a stand mixer until thoroughly combined.
03 - Add yeast mixture, lukewarm milk, eggs, melted butter, and vanilla extract to the dry ingredients. Mix on medium speed with dough hook for 5–7 minutes until smooth and slightly sticky.
04 - Transfer dough to a lightly oiled bowl, cover with plastic wrap, and let rise in a warm place for 1 to 1.5 hours until doubled in size.
05 - Punch down risen dough and turn onto a floured surface. Roll out to 1/2 inch thickness. Cut into 2-inch squares using a sharp knife or pastry cutter.
06 - Place squares on a parchment-lined baking sheet, cover lightly, and let rise for 30 minutes until puffy.
07 - Heat vegetable oil in a large, heavy-bottomed pot to 350°F. Maintain temperature throughout frying.
08 - Fry beignets in batches for 1–2 minutes per side until golden brown and puffed. Avoid overcrowding the pot to ensure even cooking.
09 - Remove beignets with a slotted spoon and drain on paper towels. Generously dust with powdered sugar while still warm and serve immediately.