Mix 1 cup sugar, butter, milk, egg, vanilla and peppermint extract thoroughly. Stir in flour, baking powder and salt. Divide dough in half. Tint one half with food color. Cover both parts dough, and refrigerate at least 4 hours.
Heat the oven to 375°F. For each cookie, roll 1 teaspoon of dough from each half back and forth on a lightly floured surface into a 4-inch rope. Place one red and one white rope side by side; press together lightly and twist. (For a smooth look, roll twisted ropes back and forth on surface several times.)
Place on ungreased cookie sheet. Curve top of cookie to form handle of cane. Bake 9 to 11 minutes or until set. Do not brown.
Mix candy and remaining sugar; immediately sprinkle over cookies. Remove from cookie sheet carefully.