Pour 1 cup of apple cider into a small saucepan over medium-high heat. Simmer the cider until it reduces to about 1/3 cup. This should take 10 to 12 minutes. This step is crucial as it concentrates the apple flavor. Once reduced, set it aside to cool completely before using it in the batter and glaze.

In a large bowl, whisk together the flour, sugar, baking powder, cinnamon, nutmeg, and salt. In a separate medium bowl, whisk together the egg, melted butter, milk, vanilla extract, and about 3 tablespoons of your cooled cider reduction. Gently fold the wet ingredients into the dry ingredients using a spatula. Mix just until combined; be careful not to overwork the dough, or the donut holes will be tough.

Heat about 2 inches of vegetable oil in a heavy-bottomed pot or Dutch oven until it reaches 350°F (175°C). Use a small cookie scoop or two spoons to drop rounded tablespoons of dough into the hot oil. Fry 4 to 5 holes at a time to avoid overcrowding. Fry for about 2 minutes per side or until they are golden brown and cooked through. Remove with a slotted spoon and drain on paper towels.

While the donut holes are still slightly warm, prepare the glaze by whisking the powdered sugar with the remaining cider reduction and cinnamon. If the glaze is too thick, add a teaspoon of milk. Dip each donut hole into the glaze, turning to coat completely. Place them on a wire rack to allow the excess glaze to drip off and set before serving.