Preheat the oven to 350 degrees Fahrenheit and line a baking sheet with parchment paper.
Finely grate the zucchini and squeeze as much liquid out as you can. You can do this one of 2 ways: 1. Place the grated zucchini into a clean linen kitchen towel. Bring the sides up and squeeze out the liquid.2. Place the zucchini into a mesh strainer set over a bowl and press firmly with your fingers until all the liquid is removed. Whisk together the flour, baking powder and salt in a medium bowl and set aside.
Place the sugars and the lemon zest in the bowl of an electric mixer and mix on medium for about 1 minute.
Add the butter and mix on medium for 2-3 minutes until light and fluffy.
Scrape down the bowl and add the egg, beating for 1-2 minutes or until the mixture is smooth.
Add the zucchini and walnuts on low until combined.
Add the flour mixture, mixing on low until just combined.
Using a 3 tablespoons size scoop, portion out the batter onto a parchment lined baking pan leaving about 2 inches between each cookie. Cover the cookies with plastic wrap anf freeze until solid. About 30 minutes.
Skipping the freezing step will result in cookies that will spread too much and become flat and not hold a dome shape.
Bake for 18-20 minutes or until they are a light golden brown and the tops spring back when lightly pressed. Remove from the oven and place the pan on a rack to cool for 10 minutes. Place the cookies directly on the rack to cool completely.
Whisk together the powdered sugar and enough lemon juice to bring the glaze to a constancy that can be drizzled. Drizzle the glaze over the cookies.
