Heat the milk in the microwave or in a small saucepan on the stovetop until very hot but not boiling.
In the bottom of a mug, stir together the sugar and cocoa powder. Add 2 tablespoons of the hot milk to make a thick paste, then slowly stir in the remaining milk (this prevents lumps from forming).
Stir in the white chocolate chips and peppermint extract, and put the candy cane into the drink as a stirring stick. It will melt into the cocoa as you stir.