Step 1: In the bowl of a stand mixer fitted with a dough hook, gently warm the whole milk to about 105-115°F (40-46°C). Sprinkle the active dry yeast and a pinch of granulated sugar over the warm milk. Let it sit for 5-10 minutes until it becomes foamy.
Step 2: Add the remaining ¼ cup granulated sugar, melted unsalted butter, large egg, and vanilla extract to the yeast mixture. Mix on low speed until well combined. Gradually add the bread flour, one cup at a time, mixing until a shaggy dough forms. Then, add the salt.
Step 3: Increase the mixer speed to medium-low and knead the dough for about 8-10 minutes, or until it is smooth, elastic, and pulls away from the sides of the bowl. The dough should be slightly sticky but manageable.
Step 4: Lightly grease a large bowl with a little oil. Transfer the dough to the bowl, turning it once to coat. Cover the bowl tightly with plastic wrap or a clean kitchen towel. Place it in a warm, draft-free spot for 1.5 to 2 hours, or until the dough has doubled in size.
Step 5: Gently punch down the risen dough to release the air. Transfer the dough to a lightly floured surface. Using a rolling pin, roll the dough into a large rectangle, about 1/4 inch thick. Using a sharp knife or a pizza cutter, cut the dough into 2-inch squares or rectangles.
Step 6: Arrange the cut beignet squares on a lightly floured baking sheet, leaving some space between each. Cover them loosely with plastic wrap or a kitchen towel and let them rise again in a warm spot for another 30-45 minutes, or until visibly puffy.
Step 7: In a large, heavy-bottomed pot or Dutch oven, heat about 3-4 inches of vegetable oil (or peanut oil) to 350°F (175°C). Use a candy thermometer to accurately monitor the oil temperature. Carefully drop 3-4 beignets into the hot oil at a time, ensuring not to overcrowd the pot. Fry for 1-2 minutes per side, or until golden brown and puffed up.
Step 8: Using a slotted spoon or spider, carefully remove the fried beignets from the oil and transfer them to a baking sheet lined with paper towels to drain excess oil. Immediately and generously dust them with powdered sugar while they are still warm. Serve immediately.