Heat the milk and half-and-half in a saucepan on medium heat to just below the simmering point. Remove the pan from the heat and add the semi-sweet and bittersweet chocolates. When the chocolates are melted, whisk in the cocoa powder, sugar, vanilla extract, and espresso, if using, and whisk vigorously. Reheat gently and pour into cups or mugs.
Crush 4 candy canes (I place them in a plastic bag and break them up with a wooden rolling pin). Stir 1/2 tablespoon of crushed candy cane into each mug, top with a dollop of whipped cream, sprinkle with crushed candy canes, and put a whole candy cane stirrer into each mug. Serve hot.
Sweetened Whipped Cream:
Place the cream, sugar, and vanilla in the bowl of an electric mixer fitted with the whisk attachment. Whisk on medium-high speed only until the cream makes soft peaks.