Please read About Doughnuts.

Whisk together in a stand mixer (or a large bowl):

Let stand until the yeast is dissolved, about 5 minutes. Add and stir in until smooth:

Cover the bowl tightly with plastic wrap and let stand at room temperature until bubbling, about 30 minutes.

Mix in with the paddle attachment (or by hand):

Add:

Switch to the dough hook and knead on medium-low speed until the dough wraps around the hook and comes away from the sides of the bowl. (Or mix the flour in with a spoon, then knead the dough until smooth and elastic.) Cover the mixer bowl with plastic wrap. (If you kneaded the dough by hand, transfer it to a bowl and cover tightly.) Let the dough rise at room temperature until tripled in volume, 1 ½ to 2 hours.

Knead the dough very briefly to deflate it, return it to the bowl, cover tightly, and refrigerate for at least 3 hours and up to 16 hours.

Roll out the dough on a lightly floured surface to ⅜ inch thick and cut out the doughnuts. Let the doughnuts rise, uncovered, at room temperature until soft and puffy, about 1 hour.

Fry, drain, and coat or glaze as for Cake Doughnuts. Begin frying the doughnuts as soon as they have risen, or they will overrise, and the taste and texture will change for the worse.