Preheat oven to 350°F (175°C) and prepare two 8-inch cake pans with parchment paper and baking spray.
Whisk together flour, baking powder, baking soda, salt, and spices in a medium bowl.
In a large bowl, whisk together the sugars, oil, applesauce, eggs, and 1 tablespoon of vanilla until smooth.
Fold the dry ingredients into the wet ingredients until just combined, then fold in the freshly grated carrots.
Divide batter evenly and bake for 35-45 minutes. Allow to cool completely.
Beat cream cheese and butter until smooth. Gradually add powdered sugar and remaining vanilla, beating until light and fluffy.
Frost the cooled cake layers, slice, and serve.