Preheat the oven to 350℉ and prepare a doughnut pan by greasing with butter or cooking spray. Set aside.
In a small sauce pot, simmer the apple cider over medium-high heat until it is reduced to a ½ cup. This will take about 10 minutes of active simmering. Remove from heat and cool.
1 cup apple cider
In a large bowl, whisk together the flour, baking powder, ground cinnamon, light brown sugar, and white sugar until well incorporated.
2 cup flour, 2 tsp baking powder, 2 tsp cinnamon, ½ cup light brown sugar, ½ cup sugar
Add the egg, vanilla, milk, and cooled cider. Stir just until combined. Batter will be thick.
1 egg, 1 tsp vanilla extract, ½ cup whole milk
Scoop or pipe the batter into the prepared baking pan, filling each mold halfway full.
Bake for 10 minutes or until a toothpick inserted in the center comes out clean. Turn out the doughnuts onto a cooling rack.
While the doughnuts are cooling, mix the white sugar with cinnamon.
1 cup sugar, 2 tsp cinnamon
Dip each cooled doughnut in the melted butter and then into the cinnamon sugar.