Place the apple cider in a 2 quart saucepan and bring to a boil. Boil the cider until it is reduced to 1/2 cup. As the cider reduces the bubbles will get larger. Once it is reduced pour it into a heatproof measuring cup and cool to room temperature.
Once the cider is cool preheat the oven to 400°F. Lightly butter the wells of a doughnut pan with 1 tablespoon of melted butter. Set aside.
Melt 4 tablespoons of butter (for the batter) in a small saucepan until it is just melted. Don’t let it get too hot. You want it just barely warm for this recipe.
Put the flour, baking powder, salt, and 1-1/2 teaspoons of the apple pie spice into a small bowl and whisk for 30 seconds to thoroughly combine. Place the egg in a medium-sized bowl and whisk.
Add the brown sugar and whisk until the brown sugar is dissolved. Make sure there are no lumps. Add the buttermilk, w 1/4 cup of the reduced apple cider and whisk to incorporate, Continuously whisk as you slowly pour in the melted butter.
Add the dry ingredients and use a silicon spatula to gently combine by stirring from the bottom until no bits of flour remain. The batter will be lumpy, which is a good thing.
Spoon the batter into an 18-inch pastry bag. Twist the top to close it and cut an 1/2-inch opening in the other end. Pipe the batter into the 6 doughnut wells of the doughnut pan.
Bake the doughnuts for 10-12 minutes. Rotating the pan halfway through the baking. The doughnuts are done when the you press them with your finger, and they spring back. Turn the doughnuts out on a wire rack to cool completely.
While the doughnuts are still warm toss in the bowl with the sugar cinnamon mixture and allow the doughnuts to cool.