1Prepare the dough: Whisk the warm milk, yeast, and sugar together in the bowl of your stand mixer fitted with a dough hook or paddle attachment. Cover and allow to sit for 5 minutes. The mixture should be a little frothy on top after 5 minutes. If not, start over with new yeast.
2Add the eggs, butter, vanilla, nutmeg, salt, and 2 cups (245g) flour. Beat on low speed for 1 minute. Scrape down the sides of the bowl with a rubber spatula as needed. Add remaining flour and beat on medium speed until the dough comes together and pulls away from the sides of the bowl, about 2 minutes.
3Knead the dough: Keep the dough in the mixer and beat for an additional 2 minutes or knead by hand on a lightly floured surface for 2 minutes.
4Let Dough Rise: Lightly grease a large bowl with oil or nonstick spray. Place the dough in the bowl, turning it to coat all sides in the oil. Cover the bowl with aluminum foil, plastic wrap, or a clean kitchen towel. Allow the dough to rise in a relatively warm environment for 1.5-2 hours or until double in size.
5Shape the Doughnuts.
6Line 1 or 2 baking sheets with parchment paper or silicone baking mats. Place doughnuts and doughnut holes on each. (Feel free to discard doughnut holes if desired.) Loosely cover and allow to rest as you heat the oil. They will rise a bit as they rest. Place a cooling rack over another baking sheet.
7Pour oil into a large heavy-duty pot set over medium heat. Heat oil to 375°F (191°C). Add 2-3 doughnuts at a time and cook for 1 minute on each side. Carefully remove with a metal spatula or metal slotted spoon. Wear kitchen gloves if oil is splashing. Place fried doughnuts onto the prepared rack. Repeat with remaining doughnuts, then turn off heat.* (See note for doughnut holes.)
8Make the glaze: Whisk all of the glaze ingredients together. Dip each warm doughnut (don’t wait for them to cool!) into the glaze, making sure to coat both sides. Place back onto prepared rack as excess glaze drips down. After about 20 minutes, the glaze will set + harden.
9Doughnuts are best enjoyed the same day. You can store in an airtight container at room temperature or in the refrigerator for 1-2 extra days.