Preheat oven to 350 degrees. Line a mini muffin tin with cupcake liners.
In a large bowl, combine grated carrots, applesauce, oil, eggs, and vanilla.
In a medium bowl, combine flour, sugar, cocoa, baking soda, salt, nutmeg, and cinnamon.
Stir dry ingredients into wet ingredients.
Using a tablespoon, fill each muffin cup about ⅔ full.
Bake for 11-15 minutes or until a toothpick inserted in the center comes out clean.
Let cool for 10 minutes in the muffin tin on a wire rack then remove cupcakes to a wire rack to cool completely.
Frosting
In an electric mixer, cream the butter and cream cheese. Add vanilla extract. Turn the mixer to low speed and add the powdered sugar a little at a time. Once mixed in, turn up the speed and mix until creamy.
When the cupcakes are cool, pipe frosting on top using a star tip. Decorate with colorful sprinkles.
Notes
To store: Place them in an airtight container and store them in the fridge where they'll keep fresh for up to three days. To serve: Remove them from the fridge and let them set on the counter for about 30 minutes before serving to bring to room temperature.