Prepare the Oven and Zucchini : Position oven racks in the upper and lower thirds and preheat to 350°F (175°C). Line your baking sheets with parchment paper. Using a box grater, shred the zucchini onto a clean kitchen towel. Gather the towel ends and wring out as much excess moisture as possible over a sink. Set the dry zucchini shreds aside.
Create the Creamed Base : In a large bowl, combine the room-temperature butter with the granulated and light brown sugars. Using an electric hand mixer on medium-high speed, beat the mixture for about 4 minutes until it becomes pale, light, and fluffy.
Enrich the Dough : Add the extra-large egg and vanilla extract to the creamed butter mixture. Beat on medium speed for another minute, scraping down the sides of the bowl with a spatula, until everything is fully incorporated and smooth.
Integrate the Dry Elements : Add the all-purpose flour, whole-wheat flour, rolled oats, cinnamon, ginger, baking soda, and salt to the bowl. Start the mixer on its lowest speed to avoid a mess, then gradually increase to medium, mixing just until the dry ingredients are barely combined and some flour streaks remain.
Fold in Key Ingredients : Switch to a spatula. Add the squeezed zucchini and the dark chocolate chunks to the dough. Fold gently until the zucchini and chocolate are evenly distributed throughout the dough. Do not overmix.
Shape and Chill the Dough : Using a 3-tablespoon scoop, portion the dough into mounds and place them on the prepared baking sheets, leaving at least 2 inches of space between each. Gently press down on each mound with your palm to flatten it to a thickness of about ½ inch. Transfer the sheets to the refrigerator to chill for 45 minutes, which helps prevent spreading.
Bake to a Golden Finish : Place the chilled cookie sheets in the preheated oven. Bake for 18 to 22 minutes, swapping the sheets between racks and rotating them 180 degrees halfway through the bake time. The cookies are ready when they are firm and appear golden at the edges.
Cool and Set : Remove the baking sheets from the oven and allow the cookies to cool on the pans for 10 minutes; this helps them set. Afterwards, carefully transfer them to a wire rack to cool completely for at least 25 minutes before enjoying.