Reduce apple cider:

Concentrate the apple cider: In a medium saucepan, bring apple cider (optionally with a cinnamon stick) to a boil. Lower the heat to medium and simmer until it reduces to 1/4 cup, roughly 20-30 minutes. Allow to cool to room temperature, at least 15 minutes.

Preheat oven and grease pan:

Set oven to 350°F (175°C) and move rack to the middle. Grease a 6-cup donut pan lightly.

Make apple powder:

In a food processor, blender, or spice grinder, process a small handful of freeze-dried apples until they become a fine powder. Measure out 1 tablespoon for the donuts; keep any extra for coating if you wish.

Whisk dry ingredients:

In a small bowl, combine almond flour, arrowroot flour, coconut flour, tapioca flour, baking powder, cinnamon, cardamom, nutmeg, allspice, salt, and freeze-dried apple powder. Whisk together.

Mix wet ingredients:

In a medium bowl, beat the egg. Add brown sugar and granulated sugar; stir to blend. Mix in applesauce, milk, and the cooled reduced apple cider.

Combine batter and fill pan:

Add the dry mixture to the wet mixture and whisk until just combined and no dry spots remain. Divide the batter evenly among the donut pan cups (a cookie scoop is useful).

Bake donuts:

Bake for 15-17 minutes, until a toothpick inserted into the center comes out clean and the donuts begin to pull away from the pan edges. Let them rest in the pan for at least 5 minutes.

Melt butter and prepare coating:

While donuts are cooling, melt the butter. Put cinnamon sugar into a small paper bag or Ziploc bag.

Coat donuts with butter and sugar:

While donuts are still warm, work with one at a time: coat the entire donut with melted butter using a brush, then drop it into the bag with cinnamon sugar and shake to cover completely. If using a paper bag, shake over the sink to catch any sugar that falls out.

Cool donuts completely:

Let donuts cool fully before serving.