Preheat the oven to 375°F (190°C), and spray two standard-sized donut pans with cooking spray. Use a paper towel to evenly distribute the grease and dab off any excess.
In a medium bowl, whisk together the flour, baking powder, cinnamon and salt.
In a large bowl, whisk together the brown sugar, egg, vegetable oil, applesauce, cider and vanilla until smooth.
Whisk the dry ingredients into the wet ingredients until combined. Be careful not to overmix (The batter won't be completely smooth.) Fold in the chopped apples.
Spoon the batter into a ziplock bag and cut off the tip, about 1 inch. Pipe the batter into the prepared donut pans, so that each donut well is almost full. (Make sure your chopped apples are smaller than the bag opening!)
Smooth the top of each donut with the back of a spoon and bake for 8 to 10 minutes, or until the top springs backs when lightly touched.
Let the donuts cool in the pan for 5 minutes before removing them to a wire rack to cool completely.