Combine applesauce, brown sugar, and eggs in a small bowl.
In a large bowl, whisk together the dry ingredients (flour, baking soda and powder, spices). Stir in the applesauce mixture. Then add the carrot, vanilla, and pineapple.
Preheat oven to 350 F. Line muffin tin with cupcake liners and fill each cup 2/3 full – these don’t rise very much, so you can be generous. Bake for 13-15 min, until a toothpick comes out clean. Oven temps vary so check for springy tops. Turn them out onto cooling racks to cool completely before frosting.
For the frosting:
Using a stand or hand mixer, beat the butter and cream cheese together until fluffy. Add in the vanilla and powdered sugar, starting to mix on low speed until the sugar is incorporated. Then beat well until mixture is smooth with no lumps.
Fit a piping bag with a round tip and fill it with the frosting, piping onto each cupcake at a 90 degree angle in a swirl from the outside in.
Optional: sprinkle the tops of the cupcakes with cinnamon, garnish edges with walnuts.