Preheat your oven to 375°F (190°C). Prepare a cookie sheet by greasing it lightly, or line it with parchment paper.
In a medium mixing bowl, combine the all-purpose flour, baking powder, and salt. Whisk together and set aside.
In a large bowl, cream the softened unsalted butter and granulated sugar together until light and fluffy. Add the egg and grated lemon zest and continue beating until well blended.
Gradually add the flour mixture into the butter mixture, stirring just until combined. Fold in the peeled, shredded zucchini until evenly distributed.
Drop the dough by rounded tablespoonfuls onto the prepared cookie sheet, spacing them apart. Bake for 10-12 minutes, or until lightly browned. Let cool on the baking sheet for a few minutes before transferring to a wire rack.
Drizzle the prepared glaze over the cookies while they are partially cooled. Allow the glaze to set before serving.
