In a small bowl, whisk together the warm milk, yeast, and 1 tsp sugar. Let sit until bubbly, 5-10 minutes.
In a large bowl, whisk together the flour and salt. In another large bowl, whisk together melted butter, egg, and 1/4 cup sugar until smooth. Stir in yeast mixture until combined. Stir in flour mixture until a shaggy dough forms. In a stand mixer knead about 5 minutes (dough may be slightly tacky). Place the dough in a large, lightly oiled bowl. Cover and let rise in a warm place until doubled in size, about 1 hour.
Line a large, rimmed baking sheet with parchment paper.
Punch down the dough. On a lightly floured surface, roll the dough into a 3/4 in thick rectangle. Using a donut cutter or 3-inch and 1-inch round cookie cutters, cut 12 rounds and 12 holes in each round, re-rolling scraps as necessary. Place the cutouts on the prepared baking sheet as you work. Loosely cover and let rise in a warm place until doubled in size, about 40 minutes.
While the donuts are rising, place a wire rack inside a rimmed baking sheet. Fill a Dutch oven with 2- inches of oil. Heat over medium heat until a deep-fry thermometer registers 350 degrees.
Working a few at a time, fry donuts in the hot oil until golden brown, about 1 minute per side for large donuts and 1 minute , constantly stirring, for the donut holes. Transfer to the wire rack to drain.