Put the crushed candy canes on a small plate. Use a wet paper towel to wet the rims of 4 mugs and dip in the candy canes to coat.
Bring the milk and white chocolate to a simmer in a small saucepan over medium-high heat, whisking until the chocolate is melted and smooth. Remove from the heat and stir in the vodka and peppermint extract.
Divide the hot chocolate among the 4 prepared mugs, immediately top with the whipped cream and sprinkle with a little of the remaining crushed candy canes.
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