Preheat the oven to 350ºF. Line a square 9x9 baking pan with parchment paper and set aside.
In a medium bowl, whisk together your dry ingredients.
In a separate bowl, combine your eggs, milk, applesauce, apple butter, and oil. Beat these into the dry ingredients until a very thick batter begins to form. Fold in the apple.
Transfer all the batter into the pan and smooth with the back of your spoon.
Bake in the center of your warmed oven for 35 - 38 minutes. Test with a cake tester by placing it in the center of the cake. If there is nothing on the stick when you remove it, the cake is ready!
Allow the cake to cool completely before frosting.
For the frosting, beat together the cream cheese, butter and maple syrup using electric beaters or stand mixer with the whisk attachment (my preferred method). Once smooth, beat in the arrowroot powder.
Dollop the frosting on top of the cake and smooth it out with an offset spatula. Sprinkle with cinnamon if desired and enjoy!