Warm the milk to 105–110°F. Pour it into a large mixing bowl, then whisk in the sugar and yeast.
Wait 5–10 minutes until the yeast looks foamy and creamy. No foam means the yeast is dead; start over with fresh yeast.
Add eggs and vanilla to the yeast mixture and whisk to combine.
In a stand mixer bowl (or the same bowl if mixing by hand), add flour and salt. Pour in the wet mixture.
Mix with the dough hook on low until a shaggy dough forms, about 1 minute.
Add the softened butter, a little at a time, and continue mixing. Increase to medium speed and knead 6–8 minutes until the dough turns smooth, slightly tacky, and elastic. The dough should clear the sides but stick slightly to the bottom.
If mixing by hand, knead on a lightly floured surface 8–10 minutes, dusting sparingly to avoid a dense dough.
Shape the dough into a ball and place it in a lightly greased bowl. Cover tightly.
Let it rise in a warm, draft-free spot until doubled, 60–90 minutes. Finger dent test: Gently press the dough; a slow spring-back with a slight indentation means it’s ready.
Turn the dough onto a lightly floured surface. Pat or roll to 1/2-inch thickness. Keep it even for consistent frying.
Cut rounds with a 3-inch cutter and punch out the centers with a 1-inch cutter. Re-roll scraps once to avoid toughness.
Place doughnuts and holes on parchment-lined trays. Lightly cover with plastic wrap.
Let the doughnuts proof until puffy and slightly jiggly, 30–45 minutes. They should look lighter and feel airy when you lift the parchment corner.
Fill a heavy pot with oil 2–3 inches deep. Heat to 350–360°F. Keep a thermometer clipped on.
Set up a wire rack over a sheet pan for draining, plus bowls for glaze or sugar.
Lift each doughnut on its parchment square to keep its shape, then ease it into the oil and remove the paper right away.
Fry 1–2 minutes per side until deep golden with a pale ring around the center. Maintain temperature—adjust heat between batches.
Transfer to the rack to drain. Fry doughnut holes last (about 1 minute total).
For glaze: Whisk powdered sugar, milk, vanilla, and salt until smooth and pourable. Dip warm doughnuts on one or both sides. Let excess drip back into the bowl.
For cinnamon-sugar: Toss warm doughnuts generously right after frying.
For chocolate: Dip cooled doughnuts in melted chocolate and add sprinkles.
