Preheat your oven to 325 degrees F.
Using a stand mixer; place the flour, butter, sugar, and salt in the bowl, add the zest of 1 lemon and the tsp of chopped rosemary. Mix on low speed, then increase to medium and mix until the dough is no longer crumbly and just comes together.
Form the dough into a flat ball and chill for about 30 minutes. On a lightly floured surface, roll out the dough to a thickness of 1/2 inch. Cut the dough into in either circles, triangles, squares, ovals, or scalloped edge shapes using a 1 1/2" cookie cutter. Re-roll the scraps as necessary.
Arrange the cookies on a baking sheet about an inch apart and chill for 15 minutes.
Bake until a light golden brown, usually 10-15 minutes depending on your oven. Remove from oven and transfer cookies to a cooling rack.
Mix the powdered sugar, lemon juice, and the remaining lemon zest until smooth. Spoon 1/2 tsp of the glaze over each cooled cookie, spread the glaze with the back of the spoon; press 1 rosemary sprig into the center of each cookie.
