Combine the water and cinnamon stick in a medium saucepan and bring to a simmer over medium heat, 5–7 minutes, until fragrant.
Add the piloncillo and stir until fully dissolved, 3–5 minutes.
Add the Mexican chocolate and whisk until melted and smooth, 2–3 minutes.
In a separate bowl, whisk the masa harina with the whole milk until completely smooth with no dry spots.
While whisking the pot, slowly pour in the milk–masa mixture. Return to a gentle simmer over medium-low heat.
Cook, whisking frequently to prevent sticking, 10–15 minutes, until the champurrado is silky and thick enough to coat the back of a spoon.
Stir in the salt, then whisk vigorously (or spin a molinillo) for 30–60 seconds to aerate and make it lightly frothy.
Serve hot in mugs. If too thick, whisk in a splash of hot water to adjust. If any lumps remain, strain through a fine-mesh sieve back into the pot and reheat briefly.