1. Pour apple cider into a saucepan and simmer until reduced to 1/2 cup, then cool.

2. Preheat oven to 350°F and grease a 12-cavity doughnut pan.

3. Whisk flour, baking powder, baking soda, cinnamon, nutmeg, and salt in a bowl.

4. In another bowl, mix melted butter with brown sugar and granulated sugar until smooth.

5. Beat in eggs one at a time, then stir in vanilla extract, buttermilk, and reduced cider.

6. Fold dry ingredients into wet ingredients until just combined, avoiding overmixing.

7. Pipe or spoon batter into the doughnut pan cavities, filling each two thirds full.

8. Bake for 10 to 12 minutes until tops spring back and are golden.

9. Brush warm doughnuts with melted butter and roll in cinnamon sugar coating.