Place the milk and vegetable shortening in a medium saucepan over medium heat and cook just until the shortening has melted. Move to the refrigerator to cool to 110ºF, about 10 minutes.
In the bowl of a stand mixer, sprinkle the yeast over the warm water and let stand for 5 minutes.
After 5 minutes, combine the cooled milk mixture with the yeast mixture. Add the sugar, salt, nutmeg, eggs, and half of the flour, and combine with the paddle attachment on low. Increase the speed to medium and beat until well combined.
Replace the paddle attachment with the dough hook, add the remaining flour, and beat until dough pulls away from the bowl and becomes smooth, 3 to 4 minutes. Remove the dough and shape into a smooth round. Transfer to a well-oiled bowl, cover, and rise until doubled in size, about 1 hour.
On a well-floured surface, fold the dough over itself twice and roll out to 3/8-inch thick. Cut out using 2 1/2 inch doughnut cutter or pastry ring, using a 7/8-inch ring for the center hole. Scrap dough can be re-rolled twice. Set on a lightly floured half sheet pan and rise, uncovered, for 30 minutes.
Heat the oil in a deep fryer (or cast-iron Dutch oven fitted with a candy/fry thermometer) to 365ºF. Drop the doughnuts into the oil 4 at a time and cook 1 minute per side. Transfer to a wire rack placed in a paper towel-lined half sheet pan. The doughnut “hole” dough can be saved and fried after the doughnuts. Fry 10 to 12 holes at a time for 1 to 2 minutes, stirring gently with a spider. Drain and remove to a wire rack set over a paper towel-lined half sheet pan. While still warm, toss in vanilla sugar or glaze.