Place the candy canes or peppermint candies inside a large resealable plastic bag and close it securely.

Using a rolling pin or the base of a heavy skillet, pound the candies until they are crushed into small, coarse pieces. For an alternative method, pulse the candies briefly in a food processor until the desired texture is achieved.

Combine the sugar and water in a small, shallow bowl, stirring until thoroughly blended.

Dip the rim of each martini glass into the sugar-water mixture. Immediately press the moistened rim into the finely crushed candy canes, creating a light and even peppermint coating.

Fill a cocktail shaker approximately two-thirds full with ice cubes.

Pour in the peppermint vodka, coffee liqueur, and milk.

Secure the shaker and shake vigorously until the cocktail is thoroughly combined and well chilled.

Strain the chilled mixture into the prepared, candy cane-rimmed martini glasses.

Serve immediately while cold.