In a large bowl, combine butter and brown sugar together with a hand mixer on high speed for about 2 minutes. Scrape the sides of the bowl as needed. Add the eggs and vanilla and beat for another minute.

Add the flour, cinnamon, ginger, nutmeg, baking powder, salt, and cornstarch to the bowl and beat on low, just until combined. Add in the old-fashioned oats and combine on low speed. Fold in the shredded zucchini.

Cover the dough with plastic wrap and refrigerate for at least 1 hour.

Preheat oven to 350°F. Line a baking sheet with parchment paper. Set aside.

Drop cookie dough by about 2-3 Tablespoons of dough onto the baking sheet.

Bake for 11-13 minutes. Don’t overbake. The edges will start to brown first. The middle of the cookie may seem like it’s underbaked, but it will continue to bake when it’s taken out of the oven.

For the frosting, beat butter and cream cheese in a mixing bowl with electric (or hand held) mixer for 3-4 minutes, scraping down the sides of the bowl as needed. Mixture should be light and fluffy.

Add in vanilla, powdered sugar, and heavy cream. Beat an additional 3-5 minutes until frosting has doubled in size and is fully blended and fluffy. Add more cream if you want the frosting thinner.

Spread frosting over cooled cookies.