Prepare the flour. Weigh and mill the soft white wheat berries on a fine setting. In a medium bowl, whisk the freshly milled flour together with the cocoa powder, baking soda, and salt.
Shred the zucchini. Finely shred the zucchini. Do not squeeze out the moisture. Set aside.
Mix the wet ingredients. In a large mixing bowl, stir together the melted butter, granulated sugar, and vanilla extract until well combined.
Add the dry ingredients. Add the dry ingredients to the butter mixture and stir until combined. The mixture will be dry and crumbly at this point. This is normal.
Add the zucchini. Stir in the finely shredded zucchini until evenly combined. Let the mixture rest for 10–15 minutes to allow the zucchini to release its moisture and the freshly milled flour to hydrate.
Preheat the oven. While the batter rests, preheat the oven to 350°F. Grease a 9x13-inch baking pan or line it with parchment paper.
Add the chocolate chips. After the batter has rested, stir in the chocolate chips and mix until everything is well combined. Spread the batter evenly into the prepared pan.
Bake. Bake for 30–35 minutes, or until a toothpick inserted into the center comes out clean with just a few moist crumbs. Let the cake cool completely before adding the frosting.
Make the frosting. In a medium bowl, whisk together the powdered sugar and cocoa powder. Add the melted butter, milk, and vanilla extract and mix until smooth and creamy.
Frost the cake. Spread the chocolate frosting evenly over the completely cooled cake. Slice and serve.