Reduce the apple cider. Pour the apple cider into a small, heavy-bottomed saucepan. Bring to a boil over medium heat and cook, whisking occasionally, until reduced to ¼ cup (2 ounces or 57 grams). Pour into a liquid measuring cup to cool slightly.
Prep your oven. Position a rack in the center of the oven and preheat the oven to 350°F.
Mix the dry ingredients for the donut batter. In a medium bowl, whisk together the flour, baking powder, cinnamon, nutmeg, and salt.
Beat the butter and sugars for the donut batter, then add the eggs, vanilla, apple cider, and dry ingredients. In the bowl of a stand mixer fitted with the paddle attachment, combine the butter and sugars. Beat on medium-high until light, fluffy, and doubled in volume, 2 to 3 minutes, using a rubber spatula to scrape down the bottom and sides of the bowl as necessary. Reduce the mixer to low and add the egg and vanilla. Scrape down the bottom and sides of the bowl.With the mixer on low, gradually add the dry ingredients and beat until just combined. Slowly pour in the apple cider. Scrape down the bottom and sides of the bowl once more, and beat on low for an additional 30 seconds. Assemble the donuts. Use a rubber spatula to scrape the batter into a piping bag fitted with a small tip.Generously spray the cavities of a donut cake pan with cooking spray. Use the piping bag to fill each cavity of the donut pan with batter until ⅔-rds full. Bake the donuts. Bake for 12 to 15 minutes, or until a skewer inserted into the center of a donut comes out with a few crumbs attached. Cool on a wire rack for 5 minutes, then turn out onto the wire rack to assemble. Repeat with the remaining donut batter. Make the donut topping. Pour the melted butter into a shallow bowl. In another shallow bowl, whisk together the sugar and cinnamon.
Coat the donuts. Quickly dip the top and bottom of a donut in the bowl of butter to coat it completely. Immediately place the donut in the bowl of sugar and cinnamon and toss until coated completely. Repeat with the remaining donuts.
Serve and store. Serve warm or at room temperature. The donuts can be stored at room temperature in an airtight container for up to 3 days.