Basic Yeast Donuts (with Many Variations)

Directions

1. In a small pot, heat the milk, water, and butter until the butter has melted and the mixture is warm (around 115°F/45°C). Remove the pot from heat.
2. In the bowl of an electric mixer fitted with the dough hook attachment, mix the flour, nutmeg, cinnamon, salt, sugar, and yeast to combine. Add the warm milk mixture and the eggs and mix on low speed to combine, 3 minutes. Raise speed to medium, and mix using a dough hook, mix until the dough begins to pull away from the edges, 3 minutes more. The dough will be sticky.
3. Transfer the finished dough to a lightly greased large bowl (you'll need enough room for the dough to rise). Cover the bowl with plastic wrap. If you're ready to make the doughnuts the same day, let it to rise until doubled in size, about 1 hour. (Alternatively, you can let it rise for 30 minutes at room temperature, then refrigerate the dough overnight so that it’s ready to make in the morning.)
4. To shape the doughnuts, roll out half of the dough on a lightly floured surface to about 1/2-inch/1 cm thick. Use a doughnut cutter to cut doughnuts (alternatively you can use two circle cutters, or even the rim of a drinking glass and the rim of a shot glass, to shape the doughnuts). I also like to cut the doughnuts into squares (about 2 inches by 2 inches) using a pizza cutter -- this is especially great for filled doughnuts.
5. Heat the oil to about 360° F. (You can use a candy/fryer thermometer, but I just do a test with one of the donut holes or a scrap piece of dough: If it rises rapidly to the surface, the oil is ready. If the donuts begin to brown too quickly, the oil is too hot.)
6. Fry the doughnuts, turning them halfway through cooking, until they're golden brown on each side. Drain on absorbent paper towels. Finish as desired. (You can glaze the donuts by dunking them, or by pouring the glaze over donuts on a rack).
7. Some finishing options:
   -Powdered: Toss in powdered sugar or cinnamon sugar.
   -Glazed: Mix 3/4 cup powdered sugar, 3 to 4 tablespoons heavy cream or milk (enough to make a runny glaze), and 1/2 teaspoon vanilla (optional).
   -Chocolate-Glazed: Mix 3/4 cup powdered sugar, 2 tablespoons dark cocoa powder, and 4 to 5 tablespoons milk or cream.
   -Chocolate-Coated: Dip doughnuts in tempered chocolate thinned with 1 to 2 tablespoons vegetable oil.
   -Fruit-Glazed: Mix 1 cup powdered sugar and 1/4 cup fruit purée.
   -Violet-Glazed: Mix 1 cup powdered sugar, 1/4 cup cream or milk, and 1 teaspoon violet extract. Garnish with candied violets.
   -Pistachio: Glaze donuts with basic glaze, then press in chopped toasted pistachios.
   -Coconut: Glaze with coconut glaze (1 cup powdered sugar, 1/4 cup coconut milk, and 1/2 teaspoon vanilla), and press in toasted coconut flakes.
   -Black and White: Make a dark chocolate ganache with 1 cup chopped dark chocolate and 1/2 cup heavy cream. Make a white chocolate ganache with 1 cup chopped white chocolate with 1/4 cup heavy cream. Glaze half the doughnut with the chocolate glaze and half with the white glaze.
   -Caramel-Glazed: Melt 1 cup of caramel candies with 1/3 cup heavy cream in the microwave in 10-second blasts until fully melted. Thin the glaze with additional milk or cream as needed to get a pourable glaze.
   -Meyer Lemon: Mix 1 cup powdered sugar with the zest and juice of 1 Meyer lemon, then add enough milk to form a pourable glaze.
   -Cinnamon Roll: Roll out the dough to 1/4-inch thick. Mix together 1 stick melted butter with 1 cup granulated sugar and 2 tablespoons ground cinnamon. Spread the mixture evenly all over the dough, then roll tightly into a cylinder. Cut into 1 inch-thick pieces, then fry until golden brown. Glaze with basic glaze.