Grease a medium bowl. Line a baking sheet with parchment paper and lightly flour it. Line a wire rack with paper towels. Set all aside.
Combine the warm water, milk, sugar, and yeast in a large bowl. Whisk to dissolve the yeast. Allow the yeast to proof for about 5 minutes or until it starts to foam.
7 ounces water, 2 tablespoons milk, 3 tablespoons granulated sugar, ¼ ounce active dry yeast
Add the remaining donut ingredients to the bowl with the yeast mixture. Stir to combine. Transfer the mixture to a stand mixer fitted with the dough hook or to a clean work surface to knead by hand. Knead on low speed (or by hand) for about 5 minutes, or until the dough is nice and smooth.
3¾ cups all-purpose flour, 1 large egg, 6 tablespoons unsalted butter, ½ teaspoon baking powder, ½ teaspoon kosher salt
Place the dough into the greased bowl, cover with plastic wrap, and let rise in a warm, draft-free place until almost doubled in size, about 1 hour.
Punch the dough down and roll out on a lightly floured surface to ½-inch thick.
Use a 3-inch cutter to cut the donuts. Place on the prepared baking sheet. Re-roll the remaining dough, cut out the remaining donuts and add them to the baking sheet.
Cover the donuts with with plastic wrap, and let proof for 15 minutes, or until puffy.
Heat 2 inches of oil in a Dutch oven to 350-360°F.
Drop 2-3 donuts at a time and fry until golden, about 2 minutes per side. Remove the donuts from the oil and place on the prepared paper towel-covered wire rack. Let cool before filling.