In a small saucepan, bring the apple cider up to a boil over high heat. Allow it to boil for about a minute then reduce the heat to a simmer.
Simmer the cider until reduced to 1/2 cup; this will take about 10-15 minutes (or longer). The cider will start to thicken a little bit and it will get slightly deeper in color. Make sure to measure out 1/2 cup when finished in case the apple cider didn't reduce quite enough.
Allow the cider to cool completely before continuing to the next step.
Preheat the oven to 375°F. Grease a donut pan with a non-stick cooking spray. I really prefer to use the nonstick spray that has flour in it for baked goods.
Whisk together the flour, baking powder, cinnamon, salt, and nutmeg in a large mixing bowl, set it aside.
Whisk together the reduced apple cider, milk, brown sugar, sugar, egg, butter, and vanilla in a medium mixing bowl.
Pour the liquid mixture into the flour mixture and stir just until combined.
Pour the batter into a large zip-top bag. Cut off about 1/2 inch off one of the bottom corners.
Pipe the batter into the pan so each hole is about 1/2 full.
Bake for 8-10 minutes until they are cooked through. You can stick a toothpick into the center of the donut, and if it comes out clean, they are done. Place the donuts onto a cooling rack.
In a medium mixing bowl, combine the sugar and cinnamon.
Brush the donuts one at a time with the melted butter (making sure to coat both sides). You could also do a real fast dip into the butter.
Dip the donut into the cinnamon mixture and shake the donut until it's well covered on all sides. Repeat this with each donut.
Serve and enjoy!
Notes
You don't have to reduce your apple cider, you can just use 1/2 cup (but. Really recommend doing it).
If you want to use apple juice you can do that but it wont give the same flavor.
These can be frozen, see my tips above.
Sifted all-purpose flour can be used in place of cake flour.
If you have a smaller donut pan, just bake these donuts in batches.
You can easily double this recipe to serve more people.