Preheat oven to 325°F. Prepare cookie sheets with parchment paper.
Using a grater, finely grate a medium sized zucchini. This should yield about 2 cups of zucchini. Place zucchini in a clean dish towel and squeeze vigorously, removing most of the liquid. You should now have about 1 cup of drained shredded zucchini. Set aside.
In a microwave-safe bowl, melt the semi-sweet chocolate chips for 30 seconds. If chips are not melted, return to microwave in 10-second increments until mostly melted. Stir frequently. When melted, set aside.
In a medium bowl, add flour, cocoa, baking soda, and salt. Sift together with a whisk. Set aside.
In a stand mixer or bowl with a hand mixer, add in butter and sugars. Mix for about a minute, or until mixture is fully combined and starts to fluff.
With a stand mixer (or hand mixer on low), add in egg and vanilla. Combine fully.
Pour in melted chocolate and mix completely.
Add flour mixture to chocolate mixture and mix on low speed until just incorporated.
Remove bowl from stand mixer (or remove hand mixer) and fold in zucchini and milk chocolate chips by hand.
With an ice cream scoop (or large spoon) drop balls of dough onto a cookie sheet, about 8 per sheet. Leave 2-inches between each ball of dough. (I did not flatten cookies at any point)
Bake for 10-12 minutes.
Remove from oven and let sit on the pan for 5-10 minutes before transferring cookies to a cooling rack.