1. Activate the yeast
In a large mixing bowl, combine the lukewarm water, sugar, and yeast. Stir gently and let it sit for about 10 minutes, until the mixture becomes foamy and fragrant.
2. Make the dough
In a separate bowl, whisk together the beaten egg, milk, and salt. Pour this into the yeast mixture and stir to combine.
Add half of the flour and mix until mostly incorporated. Then add the softened butter and the remaining flour. Stir until a shaggy dough forms.
Turn the dough out onto a lightly floured surface and knead for about 5–7 minutes, until smooth and elastic.
3. Let it rise
Lightly grease a large bowl with nonstick spray or a bit of oil. Place the dough inside, cover with plastic wrap or a clean kitchen towel, and let it rise in a warm spot for at least 2 hours, or until doubled in size.
4. Prepare for frying
Heat oil in a deep fryer or heavy-bottomed pot to 350°F (175°C). Line a plate with paper towels and set aside.
Pour about 1–2 cups of powdered sugar into a paper bag. This will be your beignet dusting station. You can also roll in a large bowl.
5. Shape the beignets
Once the dough has risen, punch it down and roll it out on a floured surface to about ¼-inch thickness. Cut into 4-inch squares using a sharp knife or pizza cutter. You may also use a cookie cutter if you want to make fun shapes.
6. Fry until golden
Carefully drop a few dough squares at a time into the hot oil. Fry, flipping occasionally, until puffed and golden brown—about 1–2 minutes per side.
Use a slotted spoon or tongs to transfer the beignets to the paper towel-lined plate to drain briefly.
7. Coat in sugar
While still warm, toss the beignets into the bag of powdered sugar. Close the bag and shake gently until each beignet is generously coated. You could also roll them around in a bowl of powdered sugar, or dust with a mesh sifter.