Preheat your oven to 350°F (175°C). Line a 9-inch square baking dish with aluminum foil, allowing the foil to overhang as handles. Grease the foil lightly.

In a large microwave-safe bowl, combine the butter and sugar. Microwave for 30 seconds, then stir with a wooden spoon. Repeat this process until the butter is melted and the mixture looks pale yellow and grainy, resembling wet sand.

Add the eggs one at a time, stirring well after each addition. Stir in the cocoa powder, salt, and vanilla extract until the mixture is shiny and fully combined.

Fold in the flour, grated zucchini, and chocolate chips by hand just until no flour streaks remain. Be careful not to overmix.

Spread the batter evenly in the prepared pan. Bake for 30 to 35 minutes, until a toothpick inserted near the center comes out mostly clean with a few moist crumbs. Allow the brownies to cool completely at room temperature.

While the brownies cool, whisk together the sugar, cocoa powder, and milk in a medium saucepan over medium-high heat. Bring to a hard boil and continue boiling for 3 minutes, stirring constantly.

Remove from heat, then add the butter, salt, and vanilla extract. Stir until the butter is melted and combined. Let the mixture cool for 5 minutes, then beat vigorously with a wooden spoon until smooth and thick.

Pour the frosting evenly over the cooled brownies. Allow the frosting to fully cool and harden before slicing into squares.