Preheat the oven to 350°F (175°C) and lightly grease a 12-cavity doughnut pan.
Pour the apple cider into a saucepan and simmer over medium heat until reduced to about ¾ cup; let it cool slightly.
In a bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In another bowl, whisk together the melted butter, sugar, egg, vanilla, buttermilk, and reduced apple cider.
Gradually add the dry ingredients to the wet ingredients and stir just until combined; do not overmix.
Spoon or pipe the batter evenly into the prepared doughnut pan, filling each cavity about three-quarters full.
Bake for 12–15 minutes, or until the doughnuts are lightly golden and a toothpick inserted into the center comes out clean.
Let the doughnuts cool in the pan for a few minutes, then transfer them to a wire rack.
Brush each doughnut with melted butter.
Combine the sugar and cinnamon in a shallow bowl, then gently coat each doughnut in the cinnamon-sugar mixture.
Serve warm or at room temperature.
