Preheat oven to 350°F. Line a 9x13-inch pan with a foil or parchment paper sling for easy removal once the brownies are done.
Combine the sugar, oil, and vanilla in a large bowl, mixing well with an electric mixer.
1 1/2 cups granulated sugar, 1/2 cup oil, 2 teaspoons vanilla extract
Add the flour, cocoa powder, soda, and salt. Mix again until a crumbly, dry mixture forms.
2 cups all-purpose flour, 1/2 cup unsweetened cocoa powder, 1 1/2 teaspoons baking soda, 1 teaspoon salt
Add the zucchini and stir or fold it in with a large wooden spoon or spatula. The batter will be quite thick and dry at first. Let it sit for 5-10 minutes so the shredded zucchini can release it's moisture, then stir again. The batter should now be thick but spreadable, somewhere between cookie dough and classic homemade brownie batter.
2 cups shredded zucchini
Spread the batter into the prepared pan, using the back of a spatula to push it into the corners. Bake for 25-30 minutes until the brownies spring back to a touch rather than leaving an indentation.
Let the brownies cool for thirty minutes. While they cool, prepare the chocolate icing by whisking powdered sugar, cocoa powder, melted butter, milk, and vanilla in a bowl until smooth. Pour over the cooled brownies, then let the brownies set for at least 30 minutes before cutting and serving. These brownies are best at room temperature or chilled from the fridge.
2 cups powdered sugar, 4 Tablespoons unsweetened cocoa powder, 1/4 cup salted butter, 1/4 cup milk, 2 teaspoons vanilla extract