Preheat the oven to 350 degrees Fahrenheit. Line a muffin pan with paper liners, and set it aside.
In a medium sized bowl, add the sugars, oil, and applesauce and stir them together until combined.
Add in the eggs, and vanilla extract and stir them into the mixture.
Dump in the flour, baking powder, baking soda, cinnamon, and nutmeg. Stir them into the mixture.
Add in the grated carrot, and fold it into the batter. Set it aside till needed.