In a large bowl, whisk warm water, sugar, and yeast. Let it sit until foamy — about 5 to 10 minutes.
Add the egg, milk, and salt. Stir in the flour and softened butter until dough comes together.
Knead on a floured surface for 5 to 7 minutes until smooth. Cover and let rise for 1 to 2 hours.
Roll dough to 1/4 inch thickness. Cut into 2-inch squares.
Heat oil to 350°F in a deep pot. Fry beignets in batches, flipping until golden, about 1–2 minutes per side.
Drain on paper towels. Dust generously with powdered sugar and serve warm (with coffee, if you're doing it right).