Pour the apple cider into a small saucepan and simmer over medium heat for 15-20 minutes, until reduced to 1/2 cup. Let it cool for 10 minutes.This concentrated cider is the secret to deep, true apple flavor in every bite.
Preheat the oven to 350°F and grease two 6-cavity donut pans well.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon and nutmeg.
In a large bowl, whisk the brown sugar, granulated sugar and melted butter. Add the reduced cider, buttermilk, applesauce and vanilla and whisk until smooth.
Add the dry ingredients and fold until just combined into a thick batter. Spoon it into a large zip-top bag, snip off one corner and pipe the batter into the donut cavities, filling each about 2/3 full.
Bake for 11-13 minutes, until the donuts spring back when lightly pressed. Cool them in the pans for 5 minutes, then gently turn them out onto a wire rack.
Mix the sugar and cinnamon in a shallow bowl. Brush each warm donut all over with the melted butter, then toss it in the cinnamon sugar until fully coated.