Prepare the Dough: In the bowl of your stand mixer, whisk together the warm milk, yeast, and 1 tablespoon of sugar. Loosely cover the bowl and let it sit for 5–10 minutes until it becomes foamy and frothy on top. If you don't have a stand mixer, you can do this in a large mixing bowl, and mix the dough with a large wooden spoon or silicone spatula. It will require a bit of arm strength. A hand mixer works, but the sticky dough often gets stuck in the beaters, so mixing by hand with a wooden spoon or silicone spatula is a better choice.

Next, add the remaining sugar, eggs, melted butter, vanilla, nutmeg, salt, and 2 cups (about 250g) of flour to the bowl. Beat on low speed for 1 minute. Scrape down the sides of the bowl with a silicone spatula as needed. Add the remaining flour and beat on medium speed until the dough comes together and pulls away from the sides of the bowl, which should take about 2 minutes. If needed, add more flour, 1 tablespoon at a time, until the dough pulls away from the sides. Be careful not to add too much flour, as the dough should remain slightly sticky.

Knead the Dough: Keep the dough in the mixer and knead on low speed for an additional 5-7 minutes, or knead by hand on a lightly floured surface for 5-7 minutes. (If you're new to bread-baking, my How to Knead Dough video tutorial can help guide you through this step.) If the dough becomes too sticky during kneading, sprinkle 1 teaspoon of flour at a time on the dough or work surface to make a soft, slightly tacky dough. Do not add more flour than needed, as you don't want the dough to become dry. After kneading, the dough should still feel soft. To check, poke it with your finger—if it slowly bounces back, your dough is ready to rise. You can also perform the "windowpane test" to check the dough’s readiness: tear off a small piece of dough (roughly the size of a golf ball) and gently stretch it until it’s thin enough to allow light to pass through it. Hold it up to a light source. If light passes through without the dough tearing, it’s ready to rise. If not, continue kneading until the dough passes the test.

Let Dough Rise: Lightly grease a large bowl with oil or nonstick spray. Place the dough in the bowl, turning it to coat all sides with oil. Cover the bowl and allow the dough to rise in a warm environment for 1.5–2 hours, or until it has doubled in size. (For a slight reduction in rise time, refer to my answer to "Where Should Dough Rise?" in my Baking with Yeast Guide.)

Shape the Doughnuts: Once the dough has risen, punch it down to release the air. Remove the dough from the bowl and transfer it onto a lightly floured surface. Roll the dough out to a thickness of about 1/2 inch. Using a 3–3.5-inch donut cutter, cut the dough into donut shapes. Re-roll the scraps and cut out more donuts. If you don't have a donut cutter, use 1 large and 1 smaller round cookie cutter (the large one should be about 3 and 1/2 inches).

Line 1 or 2 baking sheets with parchment paper or silicone baking mats. Place the doughnuts and doughnut holes on the sheets. Loosely cover them and let them rest for 30 minutes while you heat the oil. The dough will rise a little during this resting period.

Heat the Oil: Pour the oil into a large, heavy-duty pot fitted with an oil thermometer. Turn the heat to medium and bring the oil to 375°F (191°C). Add 2–3 doughnuts at a time and fry for 1 minute on each side. Carefully remove the donuts with a metal slotted spatula or metal slotted spoon. If the oil temperature begins to rise too much, lower the heat to maintain 375°F (191°C). Wear kitchen gloves if necessary to protect yourself from oil splashes. Place the fried donuts onto a prepared cooling rack. Repeat with the remaining donuts and turn off the heat when finished. For doughnut holes, see the note.

Make the Glaze: In a bowl, whisk together all the glaze ingredients. Dip each warm donut (don’t wait for them to cool) into the glaze, making sure to coat both sides. Place the donuts back onto the prepared rack to let the excess glaze drip off. Allow the glaze to set for about 20 minutes.

Storage: Doughnuts are best enjoyed the same day. Leftover doughnuts can be stored in an airtight container at room temperature or in the refrigerator for 1-2 days.