Preheat oven to 350. In a medium-large bowl combine zucchini, greek yogurt, egg, sugars, and vanilla and mix well.
In another bowl whisk together flour, baking powder, salt, and cinnamon. Add dry ingredients to wet ingredients and mix until everything is incorporated.
Drop spoonfuls of batter onto a baking sheet, at least 2 inches apart. Bake in preheated oven for 12-15 minutes. Allow to cool at least 5 minutes on the baking sheet before transferring to a cooling rack to finish cooling completely.
To make the frosting, melt butter in a medium sauce pan. When butter is melted, continue to whisk over medium heat until butter turns from yellow to a golden color. Remove from heat. In a medium bowl cream together butter, cream cheese, powdered sugar, vanilla, and milk until smooth and spreadable. Spread on cooled cookies.