Heat your oven to 350 degrees F and line two baking sheets with parchment paper. Grate the zucchini on the large holes of a box grater, then wrap it in a clean kitchen towel and squeeze firmly over the sink until no more liquid comes out. You want the shreds as dry as possible. Set aside.

Whisk together the flour, baking soda, baking powder, and salt in a medium bowl. Set aside. Place the freeze-dried strawberries in a zip-top bag and crush them with a rolling pin until they are broken into small crumbles and some fine powder. You do not want big whole pieces, which can create hollow spots in the baked cookie.

In a large bowl, beat the softened butter, granulated sugar, and 1/4 cup powdered sugar together on medium speed for about 2 minutes until light and creamy. Add the eggs one at a time, beating after each addition. Mix in the 2 teaspoons of vanilla extract.

Add the flour mixture to the butter mixture and stir gently with a rubber spatula until just combined. Fold in the squeezed zucchini and the crushed freeze-dried strawberries. The dough will be soft and slightly sticky, which is normal.

Scoop the dough using a 1 1/2 tablespoon cookie scoop, spacing the mounds about 2 inches apart on the prepared baking sheets. Bake one sheet at a time on the center rack for 11 to 12 minutes, until the edges are set and just barely golden. The centers may look slightly underdone, and that is fine. They firm up as they cool.

Let cookies rest on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely before glazing. Do not glaze warm cookies or the glaze will slide right off.

Make the glaze by whisking together 1 1/2 cups powdered sugar, 3 tablespoons milk, and 1/2 teaspoon vanilla until smooth. It should drizzle off the whisk in a slow ribbon. If it is too thick, add milk half a teaspoon at a time. Spoon or drizzle the glaze over each cooled cookie. Let the glaze set for about 15 minutes at room temperature before stacking or storing.