Combine 1 cup butter, sugar and egg in large bowl. Beat at medium speed until creamy. Add all remaining cookie ingredients; beat at low speed until well mixed. Divide dough into thirds; wrap in plastic food wrap. Refrigerate at least 2 hours or until firm.
Heat oven to 400°F. Roll out dough on lightly floured surface, one-third at a time (keeping remaining dough refrigerated), to 1/8-inch thickness. Cut with 2-inch cookie cutters. Place 1 inch apart onto ungreased cookie sheets.
Bake 6-10 minutes or until edges are lightly browned. Cool 1 minute on cookie sheet; remove to cooling rack. Cool completely.
Combine powdered sugar, 1/3 cup butter and 1 teaspoon vanilla in small bowl. Beat at low speed, adding enough milk for desired spreading consistency. Tint frosting with food color, if desired. Frost and decorate cooled cookies as desired. .
Spread frosting on top of one cookie; top with another cookie.
Chocolate Dipped Cookies:
Melt 2 cups real semi-sweet chocolate chips and 1 tablespoon shortening in 1-quart saucepan over low heat
, stirring constantly, until melted (4 to 6 minutes). Dip half of each cookie into chocolate. Place on waxed paper-lined cookie sheet; refrigerate until chocolate is firm.
: Drizzle cookies with melted chocolate as directed above.
: For easy rolling, use a lightly floured pastry cloth and stockinet (rolling pin cover), which can be purchased at department stores or specialty food stores.
: For easy decorating, spoon frosting into resealable plastic food bag. Cut off tip of one corner to create a decorating tip.
: After frosting has set, place sheet of waxed paper between cookies. Store in airtight container up to 5 days or freeze unfrosted cookies in airtight container up to 3 months.