Cookies: Cream butter and sugars in a large mixing bowl. Beat in egg and vanilla.
Whisk together flour, baking powder, soda, salt, and cinnamon. Stir into creamed mixture. (Dough will be very stiff.)
Add the grated zucchini and stir till well combined.
Drop by spoonfuls onto well greased or parchment lined cookie sheets. Bake at 375° for about 12 minutes or till bottoms are golden brown and tops are set.
Cool on pan for a couple minutes. then remove to cooling racks. Cool completely, then frost.
Frosting: Beat cream cheese and butter till smooth and creamy. Add the powdered sugar and vanilla and beat till smooth. Frost cookies.