Preheat the oven to 350°F. Spray a 9×13-inch baking dish with cooking spray.
In a medium bowl, whisk together the flour, cinnamon, baking powder, baking soda, salt, ginger, and nutmeg.
In a large bowl, whisk the eggs until smooth. Add the granulated sugar, brown sugar, oil, applesauce, and vanilla, and whisk until combined. Add the dry ingredients to the wet ingredients and mix until no flour streaks remain.
Pour the batter into the prepared baking dish and bake for 28–30 minutes, until a toothpick inserted into the center comes out clean. Let the cake cool completely before adding the topping.
To make the cream cheese topping, place the cream cheese in a large bowl and beat with a hand mixer on medium-high speed until smooth and fluffy. Add the powdered sugar, cinnamon, and vanilla and beat until smooth. Spread evenly over the cooled cake. If using, add small spoonfuls of apple butter on top and use the back of a spoon to gently swirl it into the cream cheese topping.