Cook milk in a heavy saucepan over medium heat until warm (do not boil). Add 7 crushed peppermint candies, stirring until candies dissolve. Add white chocolate, stirring with a whisk until chocolate melts. Cook until tiny bubbles form around edge of pan (do not boil), stirring constantly with a whisk. Remove from heat; stir in extract.
Coarsely crush 2 peppermint candies; set aside. Finely crush remaining 2 peppermint candies. Coat rims of 6 mugs with corn syrup. Place 2 finely crushed peppermint candies in a saucer; dip rim of each mug in candies to coat.
Pour 2/3 cup hot chocolate into each mug; top each serving with 1 tablespoon whipped topping. Sprinkle 2 coarsely crushed candies evenly over topping.