Preheat your oven to 375°F (190°C). Prepare a cookie sheet by greasing it lightly, or line it with parchment paper if you prefer easy cleanup.

In a medium mixing bowl, combine the all-purpose flour, baking powder, and salt. Whisk these together well to ensure the leavening agents are evenly distributed, then set the bowl aside for later use.

In a large bowl, cream the softened unsalted butter and granulated sugar together until light and fluffy. Add the egg and grated lemon zest, beating until well blended.

Gradually add the flour mixture into the butter mixture, then fold in the shredded zucchini until evenly distributed.

Drop rounded tablespoonfuls of dough onto the prepared sheet and bake for 10-12 minutes until lightly browned. Cool on a rack.

Drizzle the lemon glaze over the cooled cookies and allow it to set before serving.