Preheat your oven to 350 degrees. Line two muffin pans with paper liners or flour two 8 inch cake pans.
Whisk together the flour, baking soda, baking powder, salt, cinnamon, and nutmeg. Add the eggs, oil, applesauce, sugars, and vanilla extracts. Stir until smooth. Fold in carrots.
Divide between your pans. Bake 30 minutes or until a toothpick comes out clean.
Fill cupcake liners ¾ full. Bake 14-16 minutes or until a toothpick comes out clean.