Add the milk, cocoa powder, and sugar to a small saucepan over medium heat. Whisk steadily as it warms, breaking up every lump of cocoa, until the mixture is steaming, smooth, and slightly thickened, about 3 to 4 minutes. Do not let it boil. Remove from the heat and stir in the peppermint extract.
Divide the hot espresso between two mugs, then pour the chocolate milk over it, stirring to combine. If you like foam, froth a little extra milk and spoon it over the top. Finish with a generous swirl of whipped cream and a sprinkle of crushed candy cane. The candy slowly melts into the cream and adds little bursts of peppermint crunch.
For an iced peppermint mocha, let the chocolate mixture cool, pour it over ice with cold brew, and top with cold foam. It is just as festive, and in some climates December calls for ice anyway.
If you are making these for guests, set up a small topping bar: whipped cream, crushed candy canes, chocolate shavings, and a shaker of cocoa. People love building their own, and it takes the pressure off the host. A milk frother makes a nicer cap of foam, but whisking the chocolate milk vigorously right before pouring gives you a decent head too.