Preheat oven to 375 degrees. Line baking sheets with parchment paper and set aside.
Cream butter, shortening and sugar together until combined. Stir in eggs and vanilla. Scrape sides and stir again briefly. Add in all dry ingredients and mix until incorporated. Stir in lemon zest, zucchini and poppy seeds by hand. Scoop onto prepared baking sheets and bake 8 minutes or so until bottoms are lightly golden. Cool 5-10 minutes before transferring to a cooling rack.
For the glaze, stir powdered sugar into the lemon juice bit by bit until you form a thick glaze, about the same viscosity as white school glue. Drizzle over cookies and set set a few minutes before diving in. Or, pay no attention to me and snarf 'em down!