Preheat oven to 325 degrees. Spray a 9 X 13-inch baking pan with nonstick cooking spray. Set aside.
In a medium bowl, whisk together flour, 1/2 cup cocoa, baking soda, and salt. Set aside.
In a medium mixing bowl, using an electric mixer, mix together the granulated sugar, 1/2 cup butter, and oil until well combined. Add eggs, one at a time, mixing well after each addition. Stir in vanilla. Mix in dry ingredients, alternately with buttermilk in 3 additions, beginning and ending with dry ingredients. Stir in zucchini and chocolate chips. Pour batter into the prepared baking pan. Even top of the batter with a small offset spatula.
Place baking pan in preheated oven and bake for 45-50 minutes, or until the toothpick inserted in the center comes out clean. Allow cooling completely.
Meanwhile, in a medium bowl, beat 3/4 cup butter until creamy. Add powdered sugar, and 1/2 cup cocoa together until well combined. Add 2 Tablespoons of milk and beat on medium speed for 2-3 minutes. Add milk as needed, 1 tablespoon at a time until desired consistency is reached.