Heat the oven to 350°F (180°C). Line a baking sheet(s) with parchment paper; set aside.
In a medium bowl, whisk together the flour, oats, cocoa powder, baking soda, and salt.
Use a box grater to shred the zucchini. Squeeze the shredded zucchini in a paper towel to remove excess moisture.
In a large bowl, add the zucchini, yogurt, egg, brown sugar, granulated sugar, and vanilla. Mix well to combine.
Add the flour mixture to the large bowl and stir until just combined. Stir in the chocolate chips.
Scoop heaping tablespoons of the batter onto the prepared baking sheets, spacing them about 2 inches apart.
Bake until the edges are set and the middle is still slightly soft, 12 to 15 minutes total. Let the cookies cool for 5 minutes on the baking sheets, then transfer them to cooling racks to cool completely.
Serve with milk.