Place peppermint candy in a plastic bag and cover with a towel. Pound bag with a meat mallet or rolling pin until the candy is crushed into very small pieces. Transfer crushed candy pieces to a flat plate.

Run a small wedge of lime around the rim of a martini glass until it is very well moistened. Turn the martini glass upside down and place moistened rim onto the plate of crushed candy. Turn the glass gently to ensure the entire rim is evenly coated.

Fill a cocktail shaker with ice and add vodka, crème de cacao, and peppermint schnapps.

Shake vigorously until thoroughly combined.

Strain chilled liquid into a prepared martini glass and add a candy cane to the glass if desired. Serve immediately.