Using a vegetable peeler, shave milk chocolate into curls; set aside.
Place white chocolate in a medium heat-proof bowl; set aside. Place milk and cream in a medium saucepan set over medium heat until bubbles begin to form around edges of pan, about 4 minutes. Immediately pour mixture over white chocolate. When chocolate begins to melt, stir to combine. Whisk in vanilla. Continue whisking until a light foam forms.
Serve immediately garnished with peppermint sticks, whipped cream, and chocolate curls.