In the large bowl of a stand mixer, whisk ½ cup milk and 1 tablespoon granulated sugar until combined. Sprinkle yeast over and let sit until foamy and fragrant, about 5 minutes.

Whisk yeast mixture to combine, then add egg, butter, and remaining ¾ cup milk and 3 tablespoons granulated sugar. Whisk until incorporated. Add flour and salt, then place bowl in base of stand mixer fitted with the dough hook. Beat on low speed until combined, 1 to 2 minutes. Increase speed to medium-high and beat until stretchy and elastic, 6 to 8 minutes more.

Lightly grease a large bowl with oil. Gather dough into a rough ball and place in bowl. Turn once so oiled side is facing up. Cover bowl with a clean kitchen towel or plastic wrap, making sure it doesn’t touch dough. Let rise at room temperature until doubled in size, 1 to 2 hours.

On a lightly floured surface, turn out dough. Using a rolling pin, roll out dough, dusting top with flour if it sticks, to a rectangle about 12" x 10" and ¼" thick. Using a sharp knife, cut dough into 30 (2") squares.

Into a large Dutch oven or heavy pot fitted with a deep-fry thermometer, pour oil to a depth of 3". Heat over medium-high heat until thermometer registers 350°. Place a wire rack inside a large baking sheet.

Working 4 or 5 at a time, lower dough squares into hot oil. Fry, turning occasionally and adjusting heat as needed to maintain oil temperature between 350° to 360°, until golden brown, 3 to 4 minutes per side. Using a large slotted spoon or spider, transfer beignets to prepared rack and let cool slightly. Repeat with remaining dough.

Transfer beignets to a platter. Dust with confectioners’ sugar and serve warm.