Preheat the oven to 350°F (175°C). Line a 9×9-inch baking pan with parchment paper, leaving a little overhang for easy removal.
Grate 1 medium zucchini using the fine side of a box grater, then gently squeeze or pat it with paper towels to remove excess moisture. After squeezing, you should have about 120–130g of zucchini. Set aside.
In a large bowl, whisk together 1/2 cup milk, 2 eggs, 1/4 cup Greek yogurt, 1/4 cup light olive oil, 3/4 cup cane sugar, and 2 teaspoons vanilla extract until smooth and fully combined.
Add 2/3 cup oat flour, 1/2 cup cocoa powder, 1/4 cup gluten-free flour, 1 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/2 teaspoon salt to the bowl. Mix just until combined. Avoid overmixing.
Gently fold in the squeezed zucchini and 1/4 cup mini chocolate chips until evenly distributed throughout the batter.
Spread the batter evenly into the prepared pan. Bake for 34 to 38 minutes, or until a toothpick inserted in the center comes out clean or with a few moist crumbs.
While the cake cools, beat together 6 ounces cream cheese, 3 tablespoons butter, 1 teaspoon vanilla extract, and 2 tablespoons milk until smooth. Add 1/4 cup cocoa powder, 1 cup powdered sugar, 1/2 teaspoon cinnamon, and 1/8 teaspoon salt, then mix until creamy.
Once the cake is completely cool, spread the frosting evenly over the top. Slice and serve right away, or refrigerate for about 30 minutes to let the frosting set before cutting.