In a large bowl or stand mixer, beat together the butter, brown sugar, and sugar until light and fluffy, about 1 minute. Beat in the egg and grated zucchini until just combined.
In a separate bowl, whisk together the flour, baking soda, cinnamon, and salt. Add the dry ingredients to the wet ingredients in two additions, mixing just until combined after each addition.
Chill the dough for at least 2 hours.
Preheat your oven to 375 degrees F. Line a baking sheet with parchment paper or a silicone mat.
Use a medium cookie scoop to portion out the dough onto the prepared baking sheet 2 inches apart. (The cookies will spread so give them some room.) Bake for 10 to 12 minutes or until the tops of the cookies spring back when touched.
Allow the cookies 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, alternating with 1 ounce of heavy cream between additions. Add the salt and beat on high speed for 1 to 2 minutes until the frosting is silky smooth. (Add more powdered sugar or heavy cream as needed to reach your desired consistency.)
Spread the frosting over the tops of the cooled cookies and enjoy!
Notes
You do not need to squeeze or ring out the zucchini after grating, the moisture is important to the dough.
The cookie dough can be refrigerated for up to 24 hours before baking.
These cookies will keep for 3 to 4 days in an air-tight container at room temperature.