Preheat the oven to 325°F / 165°C. Grease a 9 × 13in / 23 × 33cm baking pan.
Beat 8 tablespoons of softened unsalted butter, ½ cup of vegetable oil, 1¾ cups of sugar, 1 teaspoon of vanilla extract, 1 teaspoon of baking powder, ½ teaspoon of baking soda, and ½ teaspoon of salt until smooth.
Beat in 2 large eggs. Mix in 2½ cups of all-purpose flour alternately with ½ cup of sour cream, then mix in ¾ cup of unsweetened cocoa powder.
Fold in 3 cups of shredded zucchini and ½ cup of semisweet chocolate chips. Spread in the pan.
Bake for 40-45 minutes, until the top springs back when lightly pressed and a toothpick inserted into the center comes out clean. Cool completely.
For the icing, warm 1⅓ cups of semisweet chocolate chips with 7 tablespoons of half-and-half until the chips soften, then stir until smooth. Spread over the cake.
