First, in a small saucepan, over high heat, bring the apple cider up to a boil. Boil it for about a minute, then reduce the heat to medium-low and simmer until reduced to 1/2 cup; this will take about 15-20 minutes.
The cider will thicken a little bit and it will get slightly deeper in color. After that, let the cider cool completely before using it.
Preheat the oven to 375 degrees F.
Grease a donut pan with non-stick cooking spray and set it aside.
Next, in a large bowl whisk together the flour, baking powder, cinnamon, salt, and nutmeg, and set it aside.
In another bowl, whisk together the 1/2 cup reduced apple cider, milk, brown sugar, white sugar, egg, butter, and vanilla extract.
After that, add the flour mixture to the wet ingredients and stir just until combined.
Transfer the batter to a large zip-top bag. Cut off about 1/2 inch off one of the bottom corners and pipe the batter into the prepared donut 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.