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 evenly combined.
03 - Add yeast mixture, lukewarm milk, eggs, melted butter, and vanilla extract to the dry ingredients. Mix with dough hook on medium speed 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 1-1.5 hours until doubled in size.
05 - Punch down dough and turn onto floured surface. Roll to 1/2 inch thickness and cut into 2-inch squares using a sharp knife or pastry cutter.
06 - Place squares on parchment-lined baking sheet, cover lightly, and let rise for 30 minutes until puffy.
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.