Place 1/3 cup of the warm milk in the bowl of an electric mixer. Stir in the yeast and sugar and set aside for five minutes or so to let it proof. Stir the butter into the remaining cup of warm milk and add it to the yeast mixture. With a fork, stir in the eggs, flour, nutmeg, and salt - just until the flour is incorporated. With the dough hook attachment of your mixer beat the dough for a few minutes at medium speed. Adjust the dough texture by adding flour a few tablepsoons at a time or more milk. You want the dough to pull away from the sides of the mixing bowl and eventually become supple and smooth. Turn it out onto a floured counter-top, knead a few times (the dough should be barely sticky), and shape into a ball.
Transfer the dough to a buttered (or oiled) bowl, cover, put in a warm place, and let rise for an hour or until the dough has roughly doubled in size.
Punch down the dough and roll it out 1/2-inch thick on your floured countertop. With a 2-3 inch cookie cutter, stamp out circles in the dough . Transfer the circles to a parchment-lined baking sheet and stamp out the smaller inner circles using a smaller cutter. If you cut the inner holes out any earlier, they become distorted when you attempt to move them. Cover with a clean cloth and let rise for another 45 minutes.
Bake in a 375 degree oven until the bottoms are just golden, 8 to 10 minutes - start checking around 8. While the doughnuts are baking, place the butter in a medium bowl. Place the sugar and cinnamon in a separate bowl.
Remove the doughnuts from the oven and let cool for just a minute or two. Dip each one in the melted butter and a quick toss in the sugar bowl. Eat immediately if not sooner.
Makes 1 1/2 - 2 dozen medium doughnuts.