In a large bowl, cream butter and sugar until light and fluffy. Add eggs, one at a time, beating well after each addition. Stir in vanilla.
In another bowl, combine the flour, cocoa powder, baking powder, baking soda and salt; add to the creamed mixture alternately with first cup of applesauce, beating well after each addition. Fold in zucchini and carrots. (*My zucchini was frozen, so I drained off the excess liquid and needed all 2 cups of applesauce. See how watery your vegetables are before deciding how much applesauce to use. It should be a thick cake consistency that you can easily spoon off onto the muffin tin.)
Fill paper-lined muffin cups three-quarters full. Bake at 350° for 18-22 minutes or until a toothpick comes out clean.
Cool for 10 minutes in the pans, then transfer to wire racks to cool completely before frosting.