Mix powdered sugar, butter, vanilla, lemon zest and egg in a large mixing bowl.
Stir in flour, baking soda, cream of tartar and lavender until combined.
Place dough between 2 pieces of parchment paper and roll to about ¼" thickness then refrigerate for at least 1 hour.
Preheat oven to 375 degrees. Remove dough from refrigerator and place on floured surface. Cut into circles using a cookie cutter and place on a parchment lined cookie sheet.
Bake 7-8 minutes or until edges are slightly golden, remove from oven and cool completely on a rack.
Mix together powdered sugar, milk, vanilla and lavender. Dip tops of cookies into the glaze and return to the rack to set. Enjoy!