01 - Combine warm water and yeast in a small bowl. Let stand for 5 minutes until mixture becomes foamy, indicating the yeast is active.
02 - In a stand mixer bowl, whisk together flour, granulated sugar, and salt until well combined.
03 - Add the activated yeast mixture, lukewarm milk, eggs, melted butter, and vanilla extract to the dry ingredients. Mix using the 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 to 1.5 hours until doubled in size.
05 - Punch down the 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 the squares on a parchment-lined baking sheet, cover lightly, and let rise for another 30 minutes until puffy.
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 to maintain proper oil temperature.
09 - Remove beignets with a slotted spoon and drain on paper towels. While still warm, generously dust with powdered sugar and serve immediately.