Prepare the dough. In a stand mixer bowl, whisk together warm milk, yeast, and 1 tablespoon of sugar. Let the mixture stand for 5-10 minutes until foamy. Add the remaining sugar, eggs, butter, vanilla, nutmeg, salt, and 2 cups of flour. Beat on low speed for 1 minute. Scrape down the sides of the bowl with a rubber spatula. Add the remaining flour and beat on medium speed until the dough comes together and pulls away from the sides of the bowl, about 2 minutes. If needed, add more flour, 1 tablespoon at a time, until the dough pulls away from the sides of the bowl. Be careful not to add too much flour, though. You want a slightly sticky dough.

Knead the dough. Keep the dough in the mixer fitted with a dough hook attachment and beat on low speed for an additional 5-7 minutes. Alternatively, knead by hand on a lightly floured surface for 5-7 minutes. After kneading, the dough should be smooth and elastic but still feel a little soft. Poke it with your finger—if it slowly bounces back, your dough is ready to rise. If not, keep kneading.

Let the 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 and allow the dough to rise in a warm environment for 1.5-2 hours or until doubled in size.

Shape the doughnuts. Punch down the dough to release the air. Remove the dough from the bowl and turn it out onto a lightly floured surface. Using a rolling pin, roll the dough out until it is 1/2 inch thick. Using a 3-3.5 inch doughnut cutter, cut into doughnuts. Re-roll the scraps and cut more doughnuts. Line 1 or 2 baking sheets with parchment paper or silicone baking mats. Place the doughnuts and doughnut holes on the baking sheets. Loosely cover and allow to rest for 30 minutes as you heat the oil.

Fry the doughnuts. Pour oil into a large heavy-duty pot fitted with an oil thermometer. Turn the stove on to medium heat. Heat the oil to 375°F (191°C). Add 2-3 doughnuts at a time and cook for 1 minute on each side. Carefully remove the doughnuts with a metal slotted spatula or metal slotted spoon. Be sure to lower the stove's temperature if the oil temperature is rising; you want it to stay at 375°F (191°C). Wear kitchen gloves if the oil is splashing. Place the fried doughnuts onto a prepared rack. Repeat with the remaining doughnuts, then turn off the heat.

Make the glaze. Whisk together all the glaze ingredients.

Glaze the doughnuts. Dip each warm doughnut (don't wait for them to cool!) into the glaze, making sure to coat both sides. Place the doughnuts back onto the prepared rack, as excess glaze drips down. After about 20 minutes, the glaze will set.

Enjoy!