Arrange oven racks to divide in thirds and preheat to 375°F. Lightly grease cookie sheets.
In large bowl of stand mixer, mix powdered sugar, margarine, egg, vanilla, and lemon extract. Mix in flour, baking soda and cream of tartar. Cover and refrigerate at least 2 hours. Divide dough into halves. Roll each half 3/16 inch thick on lightly floured work surface. Cut into shapes and place on prepared cookie sheet.
If not planning to decorate with frosting when cool, sprinkle with granulated sugar or colored decorating sugar; place on lightly greased sheet. Bake until edges are light brown, 7 to 8 minutes. Remove cookies to cooling rack.
If decorating with frosting, wait until completely cool. To make frosting, in small bowl combine confectioner's sugar, water, and desired food coloring. Frosting should be of a good viscosity to squeeze through pastry bag, not too stiff or soft. Add additional sugar or water to adjust. Place in pastry bag fitted with small circular tip.