Preheat oven to 350 degrees. Line cups of a standard (12-cup) muffin tin with paper or foil liners. Set aside.
In a medium bowl, mix together flour, brown sugar, baking powder, cinnamon, and salt. Mix in nuts.
In another bowl, combine zucchini, oil, eggs, apple sauce, and vanilla; add to flour mixture, and mix just until combined (do not over-mix).
Divide batter evenly among cups. Bake until a toothpick inserted in the center of a cupcake comes out clean, 40 to 45 minutes.
Cool in tin on a wire rack for 10 minutes; turn cupcakes out, right side up, and cool completely. Meanwhile, make frosting. Using an offset spatula or butter knife, spread frosting on cupcakes. The frosted cupcakes are best eaten within 1 day.
In a medium bowl, with an electric mixer, beat butter and cream cheese until smooth. Add confectioners' sugar and vanilla extract; beat until light and fluffy. Refrigerate until ready to use, up to 1 day.
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