In a small saucepan, heat the milk over low heat to 105°F to 110°F. Do not overheat or you'll risk killing the yeast.
Transfer the milk to a standing mixer fitted with bread hook attachment. Add sugar and yeast and mix for 1 minute to dissolve. Let rest, without mixing, for 5 minutes.
Turn the mixer to low speed and add the eggs and melted butter. Slowly add the flour, in 2 to 3 batches. Once all the flour is added, increase speed to medium low for 1 minute.
The dough should be sticky to the touch, but it should pull away from the sides of the bowl. If it is not, add 1 to 2 tablespoons flour. Return speed to low and mix for a full 5 minutes. Let the dough rest in the bowl for 10 minutes.
Transfer the rested dough to a greased bowl, cover with plastic wrap and refrigerated for at least 8 hours or overnight.
Cutting the doughnuts: Transfer the dough to a lightly floured surface. Roll the dough to ½-inch thickness. Cut the dough with a 3-½ inch circular round, press down firmly without twisting.¹ Place cut doughnuts on a sheet pan lined with a floured piece of parchment paper. Reform the dough and roll out a second or third time until all the dough is cut.
Let doughnuts rise in a warm area until double is size, 45 minutes to 1 hour. This can be done by preheating the oven to 200°F, shutting the oven off and leaving it open a crack with the doughnuts inside (this will cut the rise time down to 30-45 minutes). Or if baking the doughnuts, preheat the oven to 400°F and let doughnuts rise on the stove top.
Directions for baking: Transfer the baking sheets with doughnuts to the 400°F oven and bake for 8 to 10 minutes.
Directions for frying: In a deep sided pan² bring 2 quarts of vegetable oil to 365°F to 375°F. While oil heats, prepare a flat surface or rack lined with paper towels to place the fried doughnuts. Carefully transfer a few doughnuts from the sheet pan to the hot oil - if the doughnuts rose in the oven you may need a spatula. Fry on each side for 45 seconds to 1 minute, until golden brown, removing with a slotted spoon.