Combine the warm water, 1 tablespoon sugar, and yeast in a bowl. Whisk to dissolve the yeast. Leave for 5-10 minutes until foamy.
¾ cup water, ⅓ cup granulated sugar, 2¼ teaspoons active dry yeast
Add the remaining sugar, egg, egg yolk, and vanilla.
1 large egg, 1 large egg yolk, 1 teaspoon pure vanilla extract
Add the flour, baking powder, and salt. Knead until smooth using the attachment on your mixer or by hand.
3 cups all-purpose flour, ½ teaspoon baking powder, ½ teaspoon kosher salt
Slowly incorporate the butter cubes and knead well. The dough has to be glossy and smooth.
¼ cup unsalted butter
Cover the bowl with plastic wrap and let rise in a warm, draft-free place until almost doubled in size, about 1 hour.
Cut out 10 squares of parchment paper. Punch the dough down. On a floured work surface, roll the dough out to a ½-inch thickness.
Using a doughnut cutter, cut out the doughnuts and place on the prepared squares of parchment paper. Cover lightly with plastic wrap and leave to proof for about 40 minutes, or until about doubled in size.
While the donuts proof, line a baking sheet with paper towels. Set aside.
Heat 2 inches of oil in a Dutch oven to 330-350°F. To fry donuts, drop 2-3 at a time and allow to become golden before turning to the other side, about 1-2 minutes per side. Remove donuts from the oil with a spider tool and place on the prepared baking sheet.
For the glaze, mix all the glaze ingredients until smooth. Working one at a time, dip each doughnut into the glaze. Allow the glaze to set.
¼ cup water, ½ teaspoon pure vanilla extract, 2 cups powdered sugar