Preheat the oven to 350°F. Spray a 13x9-inch pan with nonstick spray.
MAKE THE BROWNIES:
In a large bowl, use an electric mixer to combine the butter and sugar until light and fluffy. Add in the eggs, one at a time, beating well after each addition. Beat in the yogurt and vanilla.
In a separate bowl, whisk together the flour, cocoa, baking soda and salt. Gradually beat the flour mixture into the wet ingredients. Then stir in the zucchini.
Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean. Let the brownie cake cool.
PREPARE THE FROSTING AND SERVE:
In a large bowl, use an electric mixer to combine the butter and powdered sugar until light and fluffy, 5 to 7 minutes. Beat in the cocoa and vanilla. Add enough milk for the frosting to achieve spreading consistency. Spread the frosting evenly over the brownie cake. Cut into pieces to serve.