Pour the 1 cup of apple cider into a small saucepan over medium heat. Bring it to a gentle simmer and let it cook uncovered for about 10 to 12 minutes, stirring occasionally, until it has reduced to roughly 1/2 cup. It should look slightly darker and smell concentrated and apple-forward. Pour it into a small bowl and let it cool for at least 10 minutes before using. Do not skip this step.

Heat your oven to 350 degrees F. Grease a 24-cup mini muffin pan well with nonstick spray or brush with melted butter, making sure to get into the bottom edges of each cup.

In a large mixing bowl, whisk together the flour, baking powder, baking soda, salt, 1 teaspoon of the cinnamon, and the nutmeg. Add the granulated sugar and brown sugar and whisk again to combine everything evenly.

In a separate bowl or large measuring cup, stir together the cooled reduced cider, milk, melted butter, egg, and vanilla extract. Pour the wet ingredients into the dry ingredients and stir with a rubber spatula until just combined. The batter will be thick and a little lumpy, which is fine. Stop stirring as soon as you do not see dry flour.

Spoon the batter into the prepared mini muffin cups, filling each one about 3/4 full. A small cookie scoop makes this faster and neater. Bake for 12 to 14 minutes, until the tops spring back when lightly pressed and a toothpick inserted in the center comes out clean.

While the donut holes bake, stir together the 1/2 cup granulated sugar and 1 teaspoon cinnamon in a small bowl and set it next to the stove. Melt the 3 tablespoons of butter in a separate small bowl.

As soon as the donut holes are cool enough to handle, about 2 minutes out of the oven, brush or roll each one lightly in the melted butter, then roll it in the cinnamon sugar until coated on all sides. Set them on a wire rack. The coating sticks best while they are still warm, so work quickly through the whole batch.