Attach the dough hook attachment to the stand mixer, if using. In the bowl of a stand mixer, add warm milk (115 degrees F), active dry yeast, and 1 teaspoon sugar. Combine and let it sit for 10 minutes until frothy.
To the same bowl, add the eggs, melted butter, vanilla extract, salt, grated nutmeg, cinnamon, and 2 cups of the bread flour. Mix on medium speed for about 1-2 minutes. Mix in remaining bread flour, 1/2 cup at a time until a dough forms. Increase speed to medium-high, and mix until dough pulls away from the bowl and a manageable dough forms. The dough should manageable, soft, and slightly sticky. If needed, add more flour little by little.
Transfer dough to a floured surface and knead by hand for about 4-5 minutes, sprinkling more flour as necessary. At this point, the dough should be elastic, very manageable, soft, and slightly sticky. It should also spring back when poked. Do not over-knead; you can tell if you over-kneaded the dough if it feels tough and dense.
Form dough into a ball and place on a lightly oiled bowl, making sure the dough is coated with oil on all sides. Vegetable oil is fine. Cover the bowl with plastic wrap and let it rise in a warm and draft-free environment for about 1.5 hours, until doubled in size. I usually let dough rise in a warm-ish oven. You can also use the microwave method (instructions below under Notes).
After the dough has risen, punch it down and transfer to a clean surface. Using a rolling pin, roll to 1/2 inch thick.
Using a donut cutter or round cookie/biscuit cutters, cut into 12 donuts. Should make 12 donuts and 12 donut holes. Transfer them to 2 large baking sheets. Cover with plastic wrap and let them rise in a warm environment for another 20-30 minutes.
In a large Dutch oven or heavy pot, heat vegetable oil to 375 degrees F. To monitor the oil’s temperature, clip a candy thermometer to the pot (like the photo above). Cook the donuts one minute per side, about 3 donuts at a time. Transfer to a rack placed on top of a large baking sheet or something to catch any dripping oil.
Fry the donut holes after you're done with the donuts, about 6 at a time. Transfer to a rack.
While the donuts cool a bit, make the maple glaze: simply whisk all the maple glaze ingredients. Tip: Add the milk and syrup little by little to get the right consistency. The glaze should be thick enough to stay on the donuts. If too watery, add more sugar.
Dip warm doughnuts in the glaze on one side. Let them cool for the glaze to harden, about 15 minutes. Enjoy!