In a small bowl, stir the warm water with 2 teaspoons of the sugar. Sprinkle the yeast on top, stir, and let it sit for 5–10 minutes, until foamy.
In a large mixing bowl or stand mixer with the dough hook, combine flour, remaining sugar, and salt. Add the milk, egg, egg yolk, melted butter, and the activated yeast mixture. Mix until everything comes together.
Knead for 5–6 minutes in a stand mixer on medium speed (or 8–10 minutes by hand) until the dough is smooth and elastic. It should be soft but not sticky.
Cover the bowl with a clean tea towel or plastic wrap. Let the dough rise in a warm spot for 1.5 to 2 hours, or until doubled in size.
Roll out the dough on a lightly floured surface into a square about ¼ inch thick. Cut into 2-inch squares using a knife or pizza cutter. (see note)
Heat the oil in a large pot to 350 degrees F using an oil thermometer. Fry the beignets in batches for 10-20 seconds per side until golden brown and puffed. Don’t overcrowd the pot and keep an eye on the beignets, they brown fast.
Remove the beignets with a spider strainer and place them on a wire rack over paper towels to drain. Then dust with powdered sugar while still warm. Don't be shy! Beignets from New Orleans come with a snowstorm of sugar!
Notes
For extra puffiness, let the cut dough rest under a towel for 20–30 minutes while you heat the oil. This is optional, but recommended if your kitchen is cool.
Maintaining 350°F oil temperature is key. Too hot = dark outside, raw inside. Too cold = oily, heavy beignets. An oil/candy thermometer is a must-have for keeping the oil at the right temperature.