Pour the 1 cup of apple cider into a small saucepan over medium-high heat. Simmer the cider until it has reduced to exactly 1/4 cup. This should take about 10-12 minutes. Set it aside to cool slightly. This step is crucial for concentrated apple flavor.

Preheat your oven to 350°F (175°C). Lightly grease a 24-cup mini-muffin tin with non-stick cooking spray or a little bit of melted butter to ensure the donut holes pop out easily after baking.

In a large mixing bowl, whisk together the all-purpose flour, 3/4 cup sugar, baking powder, salt, 1 1/2 teaspoons cinnamon, and nutmeg. Ensure there are no large lumps in the flour for a smooth batter.

In a separate medium bowl, whisk together the reduced apple cider, egg, 2 tablespoons of melted butter, buttermilk, and vanilla extract until well combined.

Pour the wet ingredients into the dry ingredients. Use a spatula to gently fold the mixture together until just combined. Be careful not to overmix, as this can make the donut holes tough rather than fluffy.

Spoon the batter into the prepared mini-muffin cups, filling each about 3/4 full. Bake for 12 to 15 minutes, or until a toothpick inserted into the center comes out clean and the tops spring back when lightly touched.

While the donut holes are baking, melt the 1/2 cup of butter in a small bowl. In another bowl, mix the 1 cup of sugar and 1 tablespoon of cinnamon. Once the donut holes are cool enough to handle but still warm, dip each one into the melted butter and then roll it generously in the cinnamon-sugar mixture.