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For the Cookies, mix flour, baking soda, cream of tartar and nutmeg. Set aside. Beat butter and sugar in large bowl with electric mixer on medium speed until light and fluffy. Beat in egg and vanilla. Gradually stir in flour mixture until well mixed. Refrigerate dough about 1 hour or until firm.
Preheat oven to 375°F. Roll out dough on lightly floured surface (see Test Kitchen Tip) to 1/8-inch thickness. Cut out dough with shamrock cookie cutter. Place on greased baking sheets.
Bake 8 to 10 minutes or until lightly brown. Cool on baking sheets 1 minute. Remove to wire racks; cool completely.
For the Icing, mix green food color and milk in small bowl. Add confectioners' sugar, stirring until color is evenly distributed and icing is smooth. To decorate cookies, hold cookie by its edge and dip top of cookie into icing. (Or spoon the icing onto cookie using a teaspoon.) Place glazed cookies on wire rack set over foil-covered baking sheet to dry.