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 ¼ cup butter with the ¼ cup granulated sugar and ¼ cup light brown sugar. Using an electric hand mixer on medium-high speed, beat the mixture for about 2-3 minutes until combined and fluffy.
Enrich the Dough: Add the extra-large egg, unsweetened applesauce, 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.
Integrate the Dry Elements: Add the white whole-wheat flour, rolled oats, cinnamon, ginger, baking soda, and optional salt to the bowl. Start the mixer on its lowest speed, then increase to medium, mixing just until the dry ingredients are barely combined.
Fold in Key Ingredients: Switch to a spatula. Add the squeezed zucchini, the ¾ cup dark chocolate chunks, and the ½ cup chopped walnuts to the dough. Fold gently until evenly distributed. 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 to flatten it to about ½ inch. Transfer the sheets to the refrigerator to chill for at least 45 minutes. This step is crucial.
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 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 to set. Afterwards, carefully transfer them to a wire rack to cool completely.