Preheat oven to 350 degrees. Spray two 9 inch round cake pans well with non stick cooking spray. You can also coat with thin layer of butter and then coat with flour and dump excess flour.
In a large mixing bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt until well combined. Set aside
In a separate large mixing bowl, whisk together the oil, eggs, brown sugar, granulated sugar, applesauce, and vanilla until fully combined. Add the grated carrots into wet ingredients and mix until well combined.
Pour the wet ingredients into the dry ingredients and mix with a whisk or rubber spatula until just combined, making sure not to over mix the batter.
Pour the cake batter evenly between both prepared cake pans. Bake at 350 degrees for 30-35 minutes or until the tops of the cakes are set and a toothpick inserted into the center of each one comes out clean. Remove from the oven, transfer to a wire rack, and allow to cool in the pans for about 20 minutes. Once the cakes have cooled, remove from the pans and return the cakes to the wire rack to finish cooling.
Using a mixer and a large bowl beat the cream cheese until smooth. Add the butter and mix until smooth. Add in the powdered sugar and vanilla extract and continue mixing until smooth.
Frost the cake when cool