Preheat over to 350°F. Spray two 9-inch round cake pans with non stick cooking spray and set aside.
Use large mixing bowl to whisk dry ingredients: flour, baking soda, baking powder, cinnamon, ginger, nutmeg, and salt until well combined. Set aside.
Use another large mixing bowl for wet ingredients: whisk together eggs, oil, brown sugar, granulated sugar, applesauce, and vanilla extract until well combined. Fold grated carrots into the mixture until well combined.
Pour the wet ingredients into the dry ingredients using a whisk or rubber spatula until well combined. Do not over mix the batter.
Pour cake batter evenly into the two pans. Bake at 350°F for 30-35 minutes until the tops of cake are set or a toothpick inserted into the center of the cake comes out clean.
Remove from oven and allow to cool in pans for 25 minutes on a wire rack. Then remove from pans and finish cooling.