Preheat the oven to 350°F. Grease a 9 x 13-inch baking pan and line the bottom with parchment paper if desired.
Finely shred the zucchini. If it is very watery, gently squeeze out some excess moisture while leaving enough moisture behind to keep the cake tender
In a medium bowl, whisk together the flour, cocoa powder, baking soda, baking powder, and salt.
In a large bowl, whisk together the granulated sugar, brown sugar, eggs, vegetable oil, and vanilla until smooth.
Add the dry ingredients to the wet ingredients, alternating with the buttermilk. Stir gently just until combined.
Fold in the shredded zucchini and chocolate chips with a rubber spatula. Avoid overmixing.
Spread the batter evenly into the prepared pan.
Bake for 30 to 35 minutes, or until the center is set and a toothpick inserted into the middle comes out with a few moist crumbs.
Let the cake cool in the pan for 10 to 15 minutes, then transfer it to a cooling rack and cool completely.
To make the brown butter frosting, melt the butter in a small saucepan over medium heat. Continue cooking until it foams and the milk solids turn golden brown with a toasted, nutty aroma
Remove the butter from the heat and let it cool until slightly warm. Transfer it to a mixing bowl.
Gradually beat in the powdered sugar, then add the vanilla and salt. Add the milk or cream one tablespoon at a time until the frosting is smooth and spreadable.
Spread the frosting over the completely cooled cake. Slice and serve at room temperature.
