In a small sauce pan, melt the butter. Stir in the cider and set aside.
Whisk together the flour, baking powder, salt, and apple pie spice in a medium bowl. Set aside.
Whisk the sugar and egg into the cider and butter mixture until evenly combined.
Stir together the wet and dry ingredients until just combined. Don't over-mix.
Spray the donut pan with cooking spray if it's not already non-stick. Add batter to a piping bag and pipe evenly into the pan (or spoon in carefully).
Bake for approximately 7 minutes - be sure to watch carefully and test with a toothpick. Allow to cool 1 minute before removing donuts onto a cooling rack.
While the donuts are baking, make the topping. Melt the butter and place into a small bowl. In a separate small bowl, stir the cinnamon, brown sugar, and white sugar together.
Dip each donut in the butter mixture and lightly coat in the dry mixture. Place back on cooling rack.