Preheat oven to 350. In a medium bowl whisk together flour, cinnamon, baking soda, oats, and salt.
In a large bowl cream butter and sugars until light and fluffy. Add egg, vanilla, and zucchini and mix well. Add dry ingredients to wet ingredients and mix until incorporated.
Drop spoonfuls of cookie dough onto a lightly greased cookie sheet at least 2 inches apart. Bake for 12-14 minutes until golden brown. Allow to cool 10 minutes on the baking sheet before transferring to a cooling rack or sheet of parchment or wax paper to finish cooling completely.
When cookies are cooled prepare frosting. Melt butter in a small sauce pan over medium heat. After melted, continue to stir over medium heat until butter turns amber-brown. Add butter, cream cheese, powdered sugar, vanilla, and milk to a large bowl and mix until smooth. Spread onto cooled cookies. Store in airtight container at room temperature.