Preheat oven to 375 degrees F.
In a medium bowl, whisk together the flour, baking powder, salt, baking soda, and ground cinnamon. Set aside these dry ingredients.
In a separate, larger bowl, whisk together the butter, eggs, granulated sugar, buttermilk, nutmeg, and applesauce. Mix until just combined.
Stir in the grated apples, then stir in the dry ingredients from step 2, and transfer the batter to a large ziplock bag. Squeeze out all of the air and snip one corner, creating a small hole. Spray the donut hole pan with a light layer of non-stick cooking spray.
Squeeze batter into the cavities of the donut hole pan and transfer to the oven to bake for 12-14 minutes.
As the donuts are baking, combine the sugar with the ground cinnamon and melt approximately 4 tablespoons of butter in a small dish.
When the donuts have finished baking, remove them from the oven and transfer to a wire rack to cool for 5-10 minutes. Brush with the melted butter and coat in the cinnamon sugar topping. Enjoy!