To prepare cookies, beat granulated sugar and butter in a large bowl with a mixer at medium speed until well blended (about 4 minutes). Add egg, egg white, and next 3 ingredients, beating well. Lightly spoon flour into dry measuring cups, and level with a knife. Combine flour and next 4 ingredients in a bowl. Add flour mixture to sugar mixture, stirring well. Spoon dough onto plastic wrap; flatten to a 1 1/2-inch thickness. Cover tightly with plastic wrap; chill 4 hours or overnight.
Preheat oven to 400°.
Roll dough to a 15 x 12-inch rectangle on a heavily floured surface. Cut dough into 20 (3-inch) squares using a sharp knife. Place squares 2 inches apart on ungreased baking sheets. Bake at 400° for 8 minutes or until golden. Immediately remove cookies from pans using a wide spatula; cool on wire racks.
To prepare frosting, combine powdered sugar and next 3 ingredients. Stir in food coloring, if desired. Spread about 2 teaspoons frosting over each cookie with an offset spatula, or place frosting in a small zip-top plastic bag, snip a tiny hole in 1 corner, and drizzle frosting over cookies. Sprinkle with assorted sugar sprinkles, if desired.