Preheat oven to 350ºF.
In a large bowl, combine the applesauce, egg, honey, butter and vanilla extract. Stir until fully combined.
In a small bowl, combine the whole wheat flour, cinnamon, baking soda, salt, and nutmeg.
Add half of the flour mixture to the wet ingredients, stir well, then add the carrots. Combine fully before adding remaining flour mixture.
Line a muffin pan with paper or silicone liners, and then portion 1/4 cup of batter into each liner.
Bake for 22-25 minutes, until an inserted toothpick comes out clean.
Let cool in the muffin pan for 2 minutes.