1In the bowl of a stand mixer fitted with the paddle attachment, stir together 2 cups of the flour, the yeast, sugar, and salt. Add the warm milk, vanilla, and egg yolks. Mix until smooth. Add the remaining flour and the butter and mix until incorporated. Continue to mix on medium speed until the dough is soft and smooth (it will be slightly sticky). Cover the bowl with plastic wrap and set in a draft-free place to rise for 1 hour, or refrigerate for up to 12 hours.

Lightly flour 2 baking sheets. Roll out the dough on a lightly floured surface and cut with a floured donut cutter or a 3- to 3-1/2-inch biscuit cutter (for the latter, you’ll also need a 1-1/2-inch biscuit cutter to form the donut holes). Discard the dough scraps, or cut into pieces to fry separately after the donuts are done, as snacks. Place the donuts and donut holes at least 1 inch apart on the baking sheets and cover loosely with plastic wrap.

Set in a warm place to rise a second time until they are puffy and nearly double in size, about 15 to 30 minutes.

The donuts are ready when the dough springs back slowly after being touched with a fingertip (if it springs back at once, it needs more time; if it collapses, it has overproofed and you can reroll the dough and cut it once). While the donuts rise, fill a large, heavy-bottomed pot with at least 2 inches of oil (the oil should not come more than halfway up the sides of the pan). Heat over medium-high heat until a deep-frying thermometer registers 360°F.

Using a spatula, carefully lower a few donuts into the oil; do not crowd the pot. Fry for 1 to 2 minutes per side, turning a few times, until the donuts are a light golden brown and cooked through. Remove with a spider or slotted spoon, drain on a wire rack over a paper towel, and let cool slightly before glazing or tossing in sugar.

Repeat with the remaining donuts and donut holes. Be sure to keep the temperature consistent while frying. Serve the donuts as is (warm is best), tossed in powdered sugar when cool, or glazed.