Preheat oven to 350F. Grease and lightly flour a 9" x 13" baking dish. In a large bowl, mix together oil, sugar and vanilla until well blended. In another bowl, combine flour, cocoa powder, baking soda and salt; stir into the sugar mixture. Fold in zucchini and chocolate chips (mixture will be thick). Empty the mixture into prepared dish and spread evenly. Bake for 25 minutes, or until brownies spring back when touched. Remove from oven and let cool in dish.
To make the frosting, melt butter in a saucepan over low heat on the stovetop. Add cocoa powder and stir. Remove from heat. In a medium bowl, combine the confectioners' sugar, milk and vanilla. Stir in the cocoa mixture. Spread frosting over cooled brownies. Optional: Score the edge of a white chocolate bar with a carrot peeler. Sprinkle the curly white chocolate shavings over frosting. (For a delicious crunch, sprinkle chopped walnuts instead.) Cut brownies into squares or rectangles and serve.