Reduce the Apple Cider - In a medium saucepan, add the apple cider (and a cinnamon stick if you like) and bring to a boil. Reduce the heat to medium and heat until the cider has reduced to a 1/4 cup. This may take 20 to 30 minutes. Let the reduced apple cider cool to room temperature for at least 15 minutes.
Preheat the oven to 350F degrees and adjust the oven rack to the middle position. Lightly grease a 6 cup donut pan.
In a food processor, blender, or spice grinder, add roughly a small handful of freeze-dried apples. Pulse to a fine powder. Note - We need 1 tablespoon of powder for the donuts. If you have extra powder, you can add it to the cinnamon sugar to coat the baked donuts!
In a small mixing bowl, whisk together the almond flour, arrowroot flour, coconut flour, tapioca flour, baking powder, spices, salt, and freeze-dried apple powder.
In a medium mixing bowl, whisk the egg. Then add the brown sugar and granulated sugar and mix to combine. Add the applesauce, milk, and cooled apple cider and mix to combine.
Sprinkle the dry ingredients over the wet ingredients and whisk until no dry streaks remain. Spoon equal portions of batter into the prepared donut pan. Tip - Use a cookie scoop to make portioning the the batter easier and cleaner.
Bake the donuts for 15 to 17 minutes, or until a toothpick inserted in the center comes out clean, and the donuts start to pull away from the sides of the donut cups. Let the donuts cool in the pan for at least 5 minutes.
While the donuts are cooling, melt the butter. Add the cinnamon sugar to a small paper bag or Ziploc bag.
While the donuts are still warm - Working with one donut at a time, use a pastry brush to brush the melted butter all around the donut. Drop the donut into the bag with the cinnamon sugar and gently swirl it around to thoroughly coat. Repeat for the remaining donuts. Tip - If using a paper bag to coat the donuts with the cinnamon sugar, swirl over the sink as the cinnamon sugar may sneak out of the bottom of the bag.
Allow the donuts to cool completely, and Enjoy!