Preheat the oven to 350 degrees 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.
In a large bowl, whisk together the flour, baking powder, ground cinnamon, light brown sugar and white sugar until well incorporated.
Add the egg, vanilla milk and cooled cider. Stir just until combined. Batter will be thick.
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.
Dip each cooled doughnut in the melted butter and then into the cinnamon sugar.
Serve warm with chilled apple cider.