Crush the peppermint candy. Swipe the rims of two rocks glasses with a little chocolate syrup, then dip into the crushed candy.
Fill a cocktail shaker with ice. Add milk, half-and-half, chocolate syrup, cocoa powder, simple syrup, peppermint extract, and vanilla.
Shake briskly for 15 seconds until well chilled and lightly frothy.
Add fresh ice to the prepared glasses. Strain the mixture evenly into both glasses.
Top with whipped cream. Drizzle with a touch of chocolate syrup and a pinch of crushed peppermint.