In a medium saucepan, heat the milk over medium heat until it's steaming but not boiling. Stir occasionally.
Remove the saucepan from the heat. Add the white chocolate chips and granulated sugar. Whisk until the chocolate is completely melted and the mixture is smooth.
Stir in the vanilla extract and peppermint extract (if using).
Pour into mugs and serve immediately. Garnish with whipped cream and chocolate shavings if desired.
Notes
For an extra festive touch, add a candy cane to each mug.