Preheat your oven to 350 degrees F. Line a large baking sheet with parchment paper.

Shred the zucchini on the fine side of a box grater. Squeeze the shreds in a clean kitchen towel over the sink until very dry, then measure 3/4 cup. Spread the pumpkin puree onto a few layers of paper towels and blot the top with more paper towels to absorb extra moisture. Set both aside.

In a medium bowl, whisk together the flour, baking soda, baking powder, salt, and pumpkin pie spice until combined.

In a large bowl, beat the softened butter with both sugars on medium speed for about 2 minutes until fluffy and pale. Add the egg and vanilla extract and beat for another 30 seconds until smooth.

Add the blotted pumpkin puree to the butter mixture and stir until combined. It may look a little broken at this point. That is fine.

Add the flour mixture to the wet ingredients and stir with a spatula until just combined and no dry streaks remain. Fold in the squeezed zucchini. The dough will be soft and scoopable rather than stiff.

Use a cookie scoop or heaping tablespoon to drop mounds of dough about 2 inches apart on the prepared baking sheet. These cookies puff rather than spread, so no need to flatten them.

Bake for 12 to 14 minutes, until the tops look set and dry and the edges are just barely pulling away from the parchment. Do not overbake. The centers should still look slightly soft. They will set completely as they cool.

Transfer cookies to a wire rack and let them cool completely before glazing, at least 20 minutes.

To make the glaze, whisk together the sifted powdered sugar, maple syrup, and 1 tablespoon of milk in a small bowl until smooth. Add milk one teaspoon at a time until the glaze runs off a spoon in a slow, steady ribbon. Drizzle over the cooled cookies with a fork or spoon. Let the glaze set for about 10 minutes before stacking or serving.