In a large bowl or stand mixer, cream together the butter, light brown sugar, and sugar until light and fluffy, about 1 minute.
Beat in the egg and finely grated zucchini just until combined.
In a separate bowl, whisk together the flour, baking soda, cinnamon, and salt.
Add the dry ingredients to the wet ingredients in two separate additions mixing just until combined.
Chill the dough for at least 2 hours.
Preheat the oven to 375°F and line a baking sheet with parchment paper and set aside.
Use a medium cookie scoop to portion out the dough onto the parchment paper 2 inches apart. Bake for 10 to 12 minutes or until the tops of the cookies spring back when touched.
Allow to cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.
While the cookies cool, add the cream cheese to a large bowl or stand mixer and beat until smooth.
Add the powdered sugar 1 cup at a time, alternative with 1 ounce of heavy cream. Add salt and beat on high speed for 1 to 2 minutes until silky smooth. Add more powdered sugar or heavy cream to reach desired consistency.
Spread the frosting over the tops of the cooled cookies.