Microwave the chocolate in a small microwave-safe bowl in 10-second increments until melted.
Crush 1 of the mini candy canes and place in a small bowl or saucer.
Dip the rims of two martini glasses in the chocolate, then the crushed peppermint.
Add the peppermint vodka, chocolate liqueur and cooled hot chocolate to a cocktail shaker filled with ice. Shake vigorously, then strain into the rimmed glasses. Garnish with the 2 remaining mini candy canes and serve.