Combine sugar and shortening in bowl of electric mixer; beat at medium speed until well blended. Add eggs, corn syrup and vanilla; beat until well blended and fluffy.
Combine 3 cups flour, baking powder, baking soda and salt in large bowl; gradually add to shortening mixture, beating at low speed until well blended.
Divide dough into 4 equal pieces; shape each into disk. Wrap with plastic wrap. Refrigerate 1 hour or until firm.
Heat oven to 375 degrees F.
Sprinkle about 1 tablespoon flour on large sheet of waxed paper. Place disk of dough on floured paper; flatten slightly with hands. Turn dough over; cover with another large sheet of waxed paper. Roll dough to 1/4-inch thickness. Remove top sheet of waxed paper.
Cut into desired shapes with floured cookie cutter. Place 2 inches apart on ungreased baking sheet. Repeat with remaining dough.
Bake 5 to 7 minutes or until edges of cookies are lightly browned. Cool 2 minutes on baking sheet; transfer to cooling rack to cool completely.
Frosting: Combine powdered sugar and milk in small bowl; stir until smooth. Add food coloring, if desired. Stir until blended. Spread on cookies; let stand until frosting is set. Decorate as desired.