Crush the candy canes or peppermints in a food storage bag using a rolling pin. Melt the white chocolate in the microwave on a flat plate for 30 seconds at a time until melted.
Dip the rim of the glass into the melted chocolate. Immediately sprinkle the crushed peppermint candies over the chocolate coating the glass. Chill in the refrigerator until ready to serve.
Fill a cocktail shaker with ice. Pour in the RumChata, peppermint schnapps, crème de cacao, and cream. Cover the shaker and shake vigorously until chilled.
Strain into the prepared martini glass and add a mini candy cane if desired. Serve cold.