1 1/2 cups milk
1/3 cup shortening
2 envelopes (4 1/2 teaspoons) instant yeast
1/3 cup warm water (105°F)
2 beaten eggs
1/4 cup sugar
1 1/2 teaspoons salt
1 teaspoon freshly ground nutmeg
23 ounces flour, plus more for dusting surface
peanut or vegetable oil, for frying (1-1 1/2 gallons, depending on fryer)
Vanilla Glaze
1/4 cup whole milk
1 teaspoon vanilla extract
2 cups powdered sugar
Chocolate Glaze
1/2 cup unsalted butter
1/4 cup whole milk, warmed
1 tablespoon light corn syrup
2 teaspoons vanilla extract
4 ounces bittersweet chocolate, chopped
2 cups sifted powdered sugar

Place milk in medium saucepan and heat over medium heat just till warm enough to melt the shortening. Place shortening in a bowl and pour warmed milk over. Set aside.
In a small bowl, sprinkle yeast over warm water and let dissolve for 5 minutes. After 5 minutes, pour yeast mixture into large bowl of a stand mixer and add milk and shortening mixture, first making sure the milk and shortening mixture has cooled to lukewarm. Add the eggs, sugar, salt, nutmeg, and half of the flour. Using the paddle attachment, combine the ingredients on low speed till flour is incorporated and then turn the speed up to medium and beat till well combined. Add remaining flour, combining on low speed at first, and then increase speed to medium and beat well. Change to the dough hook attachment of the mixer and beat on medium speed till dough pulls away from bowl and becomes smooth, approximately 3-4 minutes. Transfer to a well-oiled bowl, cover, and let rise for 1 hour till doubled in size.
On a well-floured surface, roll out dough to 3/8 inch thick. Cut out dough using a 2 1/2 inch doughnut cutter or pastry ring and using a 7/8 inch ring for the center hole. Set on floured baking sheet, cover lightly with a tea towel, and let rise for 30 minutes.
Preheat the oil in a deep fryer or Dutch oven to 365°F. Gently place doughnuts into oil, 3-4 at a time. Cook for 1 minute per side. Transfer to a cooling rack placed in baking pan. Allow to cool for 15-20 minutes prior to glazing, if desired.

Vanilla Glaze
Combine milk and vanilla in a medium saucepan and heat over low heat till warm. Sift powdered sugar into milk mixture. Whisk slowly, till well combined, try to avoid making bubbles. Remove glaze from heat and set over a bowl of warm water. Keeping glaze warm is important. This helps to avoid glazing that flakes off when cooled. Dip doughnuts into the glaze, 1 at a time, and set on a draining rack placed in a half sheet pan for 5 minutes before serving.

Chocolate Glaze
Combine butter, milk, corn syrup, and vanilla in medium saucepan and heat over medium heat till butter is melted. Decrease heat to low, add chocolate, and whisk till melted. Turn off heat, add powdered sugar, and whisk till smooth. Place mixture over a bowl of warm water and dip the doughnuts immediately. Allow glaze to set for 30 minutes before serving.