Preheat the oven to 350F. Grease and line an 8-inch square baking pan with parchment paper.
Grate the zucchini and transfer to a few paper towels to absorb some of the water.
Add the oil, brown sugar, granulated sugar, eggs, and vanilla extract in a large bowl. Use a whisk to mix until well combined.
Add the flour, cinnamon, baking powder, salt, and baking soda. Use a rubber spatula to mix until the flour is almost mixed in.
Squeeze out some additional water from the shredded zucchini then place in the large bowl. Carefully mix until the zucchini is evenly distributed throughout the dough, making sure to not over mix.
Transfer to the 8-inch pan and bake for 30-45 minutes, until the middle is set and an inserted toothpick comes out clean.
While the cake cools, make the frosting. Add the cream cheese, cocoa powder, sugar, and vanilla to a small bowl. Use a hand mixer (or stand mixer) to mix until well combined and smooth, about 1 minute.
Add the heavy cream and mix until soft peaks form, about 1 minute.
Once the cake has completely cooled, top the cake with frosting and spread until smooth.