Place candy canes or peppermint candies in large resealable plastic bag. Close bag. Pound with rolling pin or heavy skillet until coarsely crushed. Or, process in food processor using on/off pulsing action.
Mix 3 tablespoons sugar and 2 tablespoons water in small shallow bowl until well blended. Dip rim of glass in sugar water mixture, then into crushed candies to lightly coat.
Fill cocktail shaker two-thirds full with ice. Add white chocolate liqueur, half-and-half, vanilla vodka and peppermint extract; shake until well mixed and chilled. Strain into martini glasses. Serve immediately.