Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a medium bowl, whisk together flour, baking powder, and salt.
In a large mixing bowl, cream together softened butter and granulated sugar using a mixer until light and fluffy.
Beat in the egg and grated lemon zest until just combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Gently fold in the shredded zucchini.
Drop spoonfuls of dough onto prepared baking sheets, spacing slightly.
Bake for about 10-12 minutes, until edges are lightly golden.
Let cookies cool on baking sheets for a few minutes, then transfer to a wire rack to cool completely.
While cookies cool, whisk together confectioners’ sugar, fresh lemon juice, and lemon zest in a small bowl until smooth.
Once cookies are completely cool, drizzle or spread glaze over them.
Let the glaze set before serving.