Combine 1 cup butter, sugar and 1/2 cup powdered sugar in bowl. Beat at medium speed, scraping bowl often, until creamy. Add egg and vanilla; continue beating until well mixed. Add flour, baking soda and cream of tartar; beat at low speed, scraping bowl often, until well mixed.
Divide dough in half; shape each half into a ball. Wrap each in plastic food wrap; flatten slightly. Refrigerate 2 hours or until firm.
Heat oven to 375°F.
Roll out dough on lightly floured surface, one-half at a time (keeping remaining dough refrigerated), to 1/4-inch thickness. Cut with 2- to 2 1/2-inch assorted cookie cutters. Place 2 inches apart onto ungreased cookie sheets.
Bake 7-10 minutes or until edges are very lightly browned. Remove from cookie sheets. Cool completely.
Combine powdered sugar, water, 2 tablespoons butter and corn syrup in bowl. Beat at medium speed until powdered sugar is moistened. Continue beating, gradually adding additional water if necessary, until desired glazing consistency. Tint with food color, if desired.
Glaze cooled cookies. Place onto cooling rack over waxed paper. Decorate with additional glaze, decorator sugars and candies as desired. Let stand 6 hours or overnight until glaze is hardened.