Preheat your oven to 350 degrees F. Line two baking sheets with parchment paper and set aside.

Shred the zucchini on the fine side of a box grater. Pile the shredded zucchini into a clean kitchen towel, wrap it up tightly, and squeeze out as much moisture as you can over the sink. You want it as dry as possible. Set aside.

In a large bowl, beat the softened butter and granulated sugar together with a wooden spoon or hand mixer for about 2 minutes until the mixture looks pale and creamy. Add the egg, vanilla, lemon zest, and lemon juice. Stir until everything is well combined.

Add the dried zucchini to the butter mixture and stir it in. Then sprinkle in the flour, baking powder, baking soda, and salt. Stir until a soft dough forms. It will look a little shaggy at first but will come together after about 20 strokes. Do not overmix.

Scoop tablespoon-sized balls of dough onto the prepared baking sheets, spacing them about 2 inches apart. Gently flatten each ball slightly with the palm of your hand. They will not spread much on their own.

Bake for 10 to 11 minutes until the edges are just set and the bottoms are lightly golden. The centers may still look barely done, but they will firm up as they cool. Do not overbake or they will lose their soft texture.

Let the cookies cool on the baking sheet for 3 minutes, then transfer them to a wire rack. Make the glaze while they are still slightly warm.

Whisk together the sifted powdered sugar, 2 tablespoons of lemon juice, and optional lemon zest in a small bowl until smooth. It should be thin enough to drizzle but not completely watery. If it is too thick, add lemon juice a teaspoon at a time.

Use a spoon to drizzle the glaze over the warm cookies. Let them sit at room temperature for about 15 minutes until the glaze sets into a thin crackly coating before stacking or storing.