Preheat your oven to 350°F (175°C) and grease a 9x13 baking dish with butter.
In a large bowl, mix together flour, granulated sugar, brown sugar, baking soda, and salt.
In a separate bowl, whisk cocoa powder with hot water until smooth.
Add melted butter to the cocoa mixture and whisk until combined.
Mix in sour cream, eggs, and vanilla extract until well combined.
Combine the wet and dry ingredients, then gently fold in shredded zucchini. Spread the batter evenly into the prepared pan and bake for 33-35 minutes, or until a toothpick inserted into the center comes out clean. Allow the cake to cool before frosting.