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In large bowl, beat confectioners' sugar, butter, lemon zest and rosemary until well-combined. Add flour and salt; mix until combined. Dough will be crumbly. Divide dough into 2 disks. Wrap with plastic wrap and refrigerate 4 hours or until firm.
Preheat oven to 325°F. Prepare cookie pans with parchment paper.
On floured surface, roll each dough disk out to approximately 1/4 in. thick, pressing together gently if dough cracks. Cut using 3 in. cookie cutter and place on prepared pan. Re-roll dough as necessary, chilling if it becomes too soft.
Bake 12-14 minutes or until light golden brown at edges. Cool 2 minutes on pan on cooling grid. Remove from pan; cool completely on grid.
For glaze, whisk together confectioners' sugar, lemon juice and corn syrup in medium bowl until smooth. Drizzle over cooled cookies.