01 - In a small bowl, combine the warm water, warm milk, and active dry yeast. Allow the mixture to sit undisturbed for 5 minutes until it becomes foamy, indicating the yeast is active.
02 - In a large mixing bowl, thoroughly whisk together the eggs, melted unsalted butter, granulated sugar, salt, and vanilla extract until well combined.
03 - Pour the activated yeast mixture into the wet ingredients and mix thoroughly. Gradually incorporate the all-purpose flour, stirring until a soft, shaggy dough forms.
04 - Transfer the dough onto a lightly floured surface. Knead the dough by hand for approximately 5 to 7 minutes, or until it becomes smooth and elastic in texture.
05 - Place the kneaded dough into a greased bowl, ensuring all surfaces are lightly coated. Cover the bowl with a clean kitchen towel or plastic wrap. Let the dough rise in a warm, draft-free location until it has doubled in size, which typically takes about 1 hour.
06 - Once risen, gently punch down the dough to release the air. Transfer it to a floured surface and use a rolling pin to roll it out to an even thickness of about 1/2 inch.
07 - Using a sharp knife or a pastry cutter, carefully cut the rolled dough into individual 2-inch squares.
08 - In a deep fryer or a large, heavy-bottomed pot, heat the neutral oil to a temperature of 350°F (175°C). Use a candy or deep-fry thermometer to ensure accuracy.
09 - Carefully place the dough squares into the hot oil in batches, being careful not to overcrowd the pan. Fry for approximately 2 minutes per side, or until each beignet is puffed and golden brown.
10 - Using a slotted spoon, remove the fried beignets from the oil. Transfer them to a plate lined with paper towels to drain any excess oil.
11 - While the beignets are still warm, generously dust them with powdered sugar using a fine-mesh sieve. Serve immediately for the best flavor and texture.