In a small saucepan, add the apple cider and a cinnamon stick.
Simmer over medium heat until the liquid is reduced to about ½ cup. This usually takes about 20-25 minutes. You can check how much liquid is left by transferring it into a heat-proof measuring cup. At the 15-minute mark, I begin to check how much is left every 5 minutes or so.
Reserve 2 tablespoons of cider reduction for the glaze.
Preheat oven to 350° F and spray your donut pan with non-stick cooking spray.
In a large bowl, whisk together the sifted flour, baking powder, apple pie spice, cinnamon, and salt, set aside.
In a medium bowl, add together granulated sugar, brown sugar, egg, milk, melted butter, vanilla and apple cider.
Whisk until everything is combined.
Slowly stir wet mixture into dry ingredients until smooth and completely incorporated.
Fill a piping bag with the donut batter. Snip the tip about 1½-inches from the bottom and fill each cavity in the greased donut pan about two-thirds full. If you don’t have piping bags, a resealable food-storage bag will work, also. You will have enough batter to fill a 6-cavity donut pan twice.
Bake for 10-12 minutes until the donuts are slightly browned and a toothpick inserted into the thickest part comes out clean.
Allow donuts to cool until cool enough to handle, then transfer to a wire cooling rack with foil or parchment paper underneath (you’ll need to catch the glaze drippings later). Repeat with remaining batter.
Meanwhile to make the glaze, in shallow dish add together powdered sugar, milk, reserved apple cider reduction, cinnamon, and apple pie spice.
Whisk until everything is combined and smooth.
Dip the donuts face-down into the glaze, covering the top half of the donut.
Place glazed doughnuts back onto the cooling rack, glazed side up, so the glaze can drip down the sides and through the rack. If you prefer a hardened glaze, allow it to sit and harden for about 30 minutes.