Preheat your oven to 350 degrees. Grease and line two 8 inch pans or a 9x13 pan. Set aside.
Whisk together the flour, soda, powder, salt, cinnamon, and nutmeg in a small bowl. Set aside.
Mix the eggs, oil, applesauce, sugars, and vanilla in the bowl of an electric mixer. Stir in the dry ingredients and carrots. Divide between your round pans or pour all of the batter into your 9x13.
Bake 35-40 minutes or until a toothpick comes out clean.