Preheat the oven to 400℉. Grease donut pan (s) with butter or baking spray. Set aside.
In a large mixing bowl, whisk together the flour, sugar, baking powder, baking soda, cinnamon, nutmeg and salt until combined. Set aside.
In another mixing bowl, whisk together the applesauce/apple butter, oil, milk, and vanilla until smooth and combined.
Fold the wet mixture into the dry ingredients, until just mixed and no streaks of flour remain. Fold in grated or cubed apple if desired.
Transfer the batter to a large piping bag (or a ziploc bag), snip off the corner, and pipe the batter into the donut pans. Fill each donut cavity almost to the top.
Bake until golden brown, and a toothpick inserted into the center of each donut comes out clean, about 10-12 minutes.
Remove from the oven and let cool in the pan(s) for 10 minutes before transferring to a baking rack to cool.