Preheat the oven to 350 F and line a baking sheet with parchment paper.
Whisk together the flour, baking powder, baking soda, cinnamon, and salt in one bowl.
Cream the butter with both sugars until smooth, then beat in the egg and vanilla.
Stir in the zucchini, then add the dry ingredients and mix just until combined.
Drop spoonfuls of dough onto the tray and bake until the cookies are set, about 10 to 12 minutes.
Cool the cookies completely before spreading on the cream cheese frosting.
Beat the cream cheese and butter until smooth, then mix in powdered sugar, vanilla, and a little milk if needed.
Spread frosting over the cooled cookies and serve.
