3/4 cup water
1/2 cup whole milk
1/3 cup granulated sugar
11/2 teaspoons active dry yeast
1 large egg, beaten
2 tablespoons unsalted butter, very soft
3 1/2 cups all-purpose flour, divided, plus more for kneading and cutting
1/2 teaspoon kosher salt
1 quart peanut or vegetable oil, plus more for the bowl
1 cup powdered sugar
Warm the water and milk. Place the water and milk in a large microwave-safe bowl and microwave for 1 1/2 minutes on full power. Alternatively, you can warm the milk and water in a small saucepan over medium heat for about 1 minute. The mixture should be warm but cooler than 100°F.
Proof the yeast in the warm milk mixture for 15 minutes. Whisk the sugar into the milk mixture, then sprinkle with the yeast and set aside for 15 minutes until the yeast has become foamy.
Beat in the the egg, butter, and flour. Add the egg, salt, butter, and 2 cups of the flour to the milk mixture. Beat with a wooden spoon or sturdy spatula until the mixture is thick but smooth. Add the remaining 1 1/2 cups flour and beat again until just combined. You’re looking for a thick batter-like consistency that is closer to drop biscuits than bread dough here.
Cover a work surface with flour and knead the dough. Dust a work surface with about 1/4 cup more flour and dump the dough onto it. Gently fold the dough over itself 3 to 4 times (the dough will be sticky and a little loose). Gently shape the dough into a round and place in a well-oiled bowl (you can reuse the mixing bowl if it is mostly scraped clean). Cover with a clean kitchen towel.
Let the dough rise for 1 1/2 to 2 hours or overnight. Set the bowl in a warm place to rise until doubled in size, 1 1/2 to 2 hours. Alternatively, you can cover the bowl with plastic wrap and let it rise overnight in the refrigerator for 8 to 10 hours.
Heat the oil in a large Dutch oven to 370°F. Heat the oil in a large Dutch oven fitted with a deep-fry thermometer over medium-high heat until 370ºF. Line a baking sheet with two layers of paper towels. If the dough is in the refrigerator, pull it out just before you heat the oil to give it 20 to 25 minutes to lose its chill while the oil warms up.
Roll out and cut the dough. While the oil is heating, dust a work surface with about 1/4 cup of flour and dump the dough onto it. Roll the dough out to a 17×11-inch rectangle that is about 1/4-inch thick. Use a pizza cutter or sharp knife to cut the dough into 20 to 22 (about 2-inch square) pieces. Don’t fret if some of the corner pieces are more triangular, and don’t reroll the dough.
Fry 3 to 4 pieces at a time for 2 to 3 minutes. Fry 3 to 4 squares at a time: Gently add the beignets to the oil (you can use your hands or tongs). Immediately use a large heatproof spoon to ladle the hot oil over the pieces. Fry the first side until golden brown, 1 to 2 minutes. Using the spoon or tongs, flip the beignets and fry until the second side is golden brown, 1 to 2 minutes more.
Transfer the fried beignets to the baking sheet. Transfer the pieces to the baking sheet and repeat with the remaining dough.
Dust the beignets heavily with powdered sugar just before serving. Just before serving and while still warm, use a large fine-mesh strainer to generously dust the beignets with powdered sugar.
Make ahead: These beignets are best eaten fresh, but the dough can be made the night before and proofed in the refrigerator overnight.