For Cookies: Preheat oven to 375° F. Line 2 large cookie sheets with parchment or wax paper.
In a large bowl whisk together flour, baking powder, salt, cinnamon and ginger. Set aside.
In a separate large bowl, or stand-mixer using paddle attachment, cream butter and both sugars together until light and fluffy. Mix in egg and vanilla until just combined. Slowly mix in half the flour mixture, then the zucchini and then the rest of the flour mixture.
Drop batter into 2 Tablespoon balls onto prepared cookie sheets. Bake 11 to 13 minutes only. Don't over bake. Remove from oven and cool cookies on the cookie sheets. Once cooled frost.
Chocolate Cream Cheese Frosting: In a large mixing bowl, beat cream cheese and butter together until combined. Gradually add in powdered sugar, then vanilla and cocoa powder until all combined and creamy. Frost cooled cookies with frosting and serve.
Store cookies covered in fridge for up to 3 days.