Pour the warm milk (105–110°F) into your mixing bowl, stir in 1 tablespoon of the granulated sugar, and sprinkle the yeast over the top. Let it sit for 5–10 minutes until foamy and bubbly — if it doesn't foam, the yeast is inactive and you'll need to start again with a fresh packet.
Add the remaining sugar, eggs, vanilla extract, nutmeg, salt, and softened butter to the yeast mixture, then mix on low speed. Add the flour one cup at a time until a soft, slightly sticky dough forms and pulls away from the sides of the bowl.
Knead the dough using the dough hook on medium speed for 8–10 minutes (or by hand for 12–15 minutes) until it is smooth, elastic, and springs back slowly when poked. Add flour one tablespoon at a time only if the dough remains extremely sticky after several minutes of kneading.
Place the dough in a lightly oiled bowl, turn once to coat, then cover with plastic wrap and set in a warm, draft-free spot. Let it rise for 1 to 1.5 hours until doubled in size.
Punch the dough down and roll it out on a lightly floured surface to a uniform 1/2-inch thickness. Use a 3-inch donut cutter to cut out donuts and donut holes, then transfer to parchment-lined baking sheets, re-rolling scraps as needed.
Cover the cut donuts loosely with plastic wrap and let them rise in a warm spot for 30–45 minutes until visibly puffed and pillowy. Do not skip this step — it is essential for a light, airy texture.
Pour 3–4 inches of neutral frying oil into your Dutch oven and heat over medium to medium-high heat until a deep fry thermometer reads 350–360°F (175–180°C). Set up a wire rack over paper towel-lined baking sheet for draining.
Carefully lower 2–3 donuts into the hot oil and fry for about 1 minute per side until golden brown, monitoring the thermometer and adjusting the heat to maintain 350–360°F between batches. Fry donut holes last, about 45–60 seconds per side, then transfer all to the wire rack to drain.
Whisk together the sifted powdered sugar, whole milk, vanilla extract, and pinch of salt in a medium bowl until completely smooth. Adjust consistency with additional milk (too thick) or powdered sugar (too thin) one teaspoon at a time.
While the donuts are still warm, dip each one face-down into the glaze for 2–3 seconds, let excess drip off, then flip right-side-up onto the wire rack. Allow the glaze to set for 5–10 minutes before serving; toss donut holes in cinnamon sugar while still warm if desired.