Preheat oven to 350 degrees F. Grease a donut pan with oil and set aside.
In a large mixing bowl, beat the eggs, apple juice, maple syrup, melted butter, apple sauce, vanilla, and milk together until fully combined.
In a separate bowl, mix the flour, baking soda, baking powder, salt, cinnamon and nutmeg together. Slowly mix the dry into the wet, and stir until just combined. Fold in the grated apple.
Place about 1/2 of the batter into a piping bag or zip lock bag with one corner cut open. Carefully pipe the batter into the donut pan. Bake 13-16 minutes or until the donuts are cooked through.
To make the cinnamon sugar, combine the cinnamon and sugar together in a bowl and set aside.
Let the donuts cool slightly before removing from the donut pan. Brush melted butter onto the donuts, then dip into cinnamon sugar, making sure to cover the entire donut. Enjoy immediately!