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1. In a medium bowl stir together the flour, cream of tartar, baking soda, and 1 teaspoon lemon peel. Set aside.
2. In a large mixing bowl, beat the sugars with the butter with an electric mixer on medium speed until combined. Beat on low speed until smooth and pale, about 1 minute. Add the egg, 2 tablespoons rye, 1/2 teaspoon Herbsaint, 1/2 teaspoon bitters, and vanilla. Beat on medium speed, until fluffy, 1 to 2 minutes. Beat in the dry ingredients on low speed, just until combined.
3. Divide dough in half. Wrap and chill for 1 hour or until easy to handle.
4. Preheat the oven to 375 degrees F. On a lightly floured work surface, roll out half the dough at a time to 1/4-inch thickness. Cut dough with 2-inch round cookie cutters. Arrange 2 inches apart on ungreased cookie sheets.
5. Bake cookies for 8 to 10 minutes, just until lightly brown around edges. Completely cool cookies on wire racks before frosting.Frosting:
6. Combine all of the ingredients in a medium bowl and stir until frosting is smooth and spreadable, adjusting milk amount as necessary. If desired, garnish each cookie with additional finely shredded lemon peel.