Combine the milk and a cinnamon stick in a medium saucepan and warm over medium heat until steaming, 3–5 minutes; reduce heat and simmer gently to infuse, 3–5 minutes, then fish out the stick and discard.
Add the Mexican chocolate tablet, chopped, and whisk over medium-low heat until fully dissolved and smooth, 3–5 minutes; keep below a boil to prevent scorching.
Take the pan off the heat and froth with a molinillo (rolled briskly between your palms) or a whisk until a thick, velvety foam forms, 30–60 seconds.
Pour into 4 mugs, re-frothing between pours to keep the foam even. Serve hot.