01 - In a small bowl, combine the warm water, warm milk, and active dry yeast. Allow the mixture to rest for 5 minutes, or until a foamy layer appears on the surface, indicating the yeast is active.
02 - In a large mixing bowl, whisk together the eggs, melted butter, granulated sugar, salt, and vanilla extract until thoroughly combined.
03 - Pour the activated yeast mixture into the wet ingredients and mix well. Gradually incorporate the all-purpose flour, stirring until a soft, shaggy dough begins to form.
04 - Transfer the dough to a lightly floured surface. Knead the dough by hand for approximately 5 to 7 minutes, or until it becomes smooth and elastic.
05 - Place the kneaded dough in a lightly greased bowl, turning once to coat. Cover the bowl with a clean kitchen towel or plastic wrap and allow it to rise in a warm location until its volume has doubled, which typically takes about 1 hour.
06 - Gently punch down the risen dough to release the air. Transfer it to a floured surface and roll it out evenly to an approximate thickness of 1/2 inch.
07 - Utilizing a sharp knife or a pastry cutter, divide the dough into uniform 2-inch squares.
08 - In a deep fryer or a large, heavy-bottomed pot, heat the neutral oil to 350°F.
09 - Carefully place the dough squares into the hot oil in batches, ensuring not to overcrowd the pot. Fry each side for about 2 minutes, or until the beignets are puffed and golden brown. Maintain a consistent oil temperature.
10 - Using a slotted spoon, remove the fried beignets from the oil and transfer them to a plate lined with paper towels to drain excess oil. While the beignets are still warm, generously dust them with powdered sugar.
11 - Serve the carnival-style beignets immediately for the best flavor and texture.