Preheat oven to 350 degrees. Spray donut pans, or muffin tins with nonstick cooking spray.
In a medium bowl, whisk together flour, baking powder, salt, cinnamon, and apple pie spice.
In a separate medium bowl or stand mixer, cream together butter and sugars until light and fluffy. Add eggs and apple sauce, beat until combined. Add in vanilla until combined.
Stir in the dry ingredients and apple cider, alternating back and forth between stirring. Stir just until batter comes together. Spoon or pipe into donut pans. Bake for 15-17 minutes or until a toothpick comes out clean, and donuts spring back to the touch.
Cool on a wire rack in the donut pan for about 5 minutes before removing donuts from the pan.
While the donuts are cooling, melt butter in a small bowl. In a separate shallow dish, combine the sugar and cinnamon. Brush donuts with melted butter, and dredge donuts in cinnamon sugar mixture. Enjoy immediately or at room temperature. Best enjoyed the same day!
If you do not have donut pans you can use a muffin tin, or mini muffin tin, but your baking times may vary. Be sure to check periodically with a toothpick.