In the bowl base of a stand mixer, combine warm milk, yeast, and sugar. Let it sit for about 15 minutes, until it’s foamy and alive. (If it doesn’t foam, start over — the yeast is dead.)
1 ¼ cup whole milk, warmed to 90°F (32°C), 2 ¼ teaspoon active dry yeast (1 packet), ¼ cup granulated sugar
Using a stand mixer fitted with the dough hook, beat on medium-low speed the melted butter, eggs, vanilla, and salt in the yeast mixture, about one minute. Add one third of the flour and mix to combine. Repeat with another third, then add the remaining flour gradually until a soft, slightly tacky dough forms, three to five minutes, scraping down the sides as necessary.
8 tablespoon unsalted butter, 2 large egg, 1 teaspoon vanilla extract, 1 teaspoon salt, 4 cup all-purpose flour
Increase the mixer to medium-high speed and knead for about three minutes, until the dough starts to pull away from the bowl, seven to ten minutes. Remove the dough onto a floured surface, dip your hands into flour, and gently hand-knead until smooth and elastic. Add flour to your hands as needed until the dough is smooth and does not stick to your hands or the surface.
Grease a large bowl with softened butter, transfer the dough to the bowl, cover, and let rise in a warm place until doubled in size, about one hour.
Cut two parchment sheets into thirds and line two baking sheets with parchment pieces.
Press the dough into an 11X14 rectangle, about half-inch thick. Cut with a donut cutter dipped in flour (or a three and a half-inch (eight point nine centimeter ring and one-inch (two and a half centimeter) center cutter). Transfer to a parchment-lined baking sheet, cover lightly, and let rest for 1 hour or until puffy and slightly larger.
Heat oil in a heavy pot to 350°F (176°C). Fry two to three donuts at a time, about one minute per side, until golden brown. Drain on a wire rack set over a sheet pan. Cool slightly before glazing, about five minutes.
Neutral oil for frying
Whisk each glaze in a separate bowl until thick, smooth, and spoonable. Add more sugar for thickness or more milk for smoothness. The goal: thick, shiny, frosting-style coating.
1¼ cup powdered sugar, 2 tablespoon whole milk, ½ teaspoon lemon juice, pinch salt, 1 cup powdered sugar, ¼ cup unsweetened cocoa powder, 2 tablespoon milk, pinch salt, 1½ cup powdered sugar, 2 tablespoon seedless strawberry jam, ¼ teaspoon beet juice (optional, for color), 1 tablespoon milk (adjust for consistency)
Dip the top of each donut in its glaze, twist gently, and lift off to create a smooth, even coating. Top immediately with sprinkles — chocolate on chocolate, rainbow on vanilla and strawberry.
rainbow sprinkles, chocolate sprinkles
Let the glaze set for about 15 to 30 minutes (if you can wait that long). Serve fresh!