Step 1: In a medium bowl, whisk together the masa harina with 1 cup of the milk until no lumps remain and a smooth slurry forms. Set aside.
Step 2: In a large saucepan or Dutch oven, combine the remaining 3 cups of milk, 1 cup of water, cinnamon sticks, and star anise (if using). Heat over medium heat until it just begins to simmer, stirring occasionally. Do not boil.
Step 3: Reduce the heat to low. Add the chopped Mexican chocolate and piloncillo (or brown sugar) to the simmering liquid. Stir continuously with a whisk or molinillo until the chocolate and piloncillo are completely dissolved, about 3-5 minutes.
Step 4: Slowly pour the masa harina slurry into the saucepan while continuously whisking to prevent lumps. Continue to whisk for 1-2 minutes until the mixture begins to thicken.
Step 5: Increase the heat slightly to medium-low and continue to cook, stirring frequently (every 30-60 seconds) for about 10-15 minutes, or until the champurrado reaches your desired thickness and the raw masa flavor is cooked out. It should be thick enough to coat the back of a spoon.
Step 6: Carefully remove the cinnamon sticks and star anise from the champurrado. Taste and adjust sweetness if necessary.
Step 7: Pour the hot Mexican Champurrado into mugs and serve immediately. Garnish with a sprinkle of ground cinnamon, if desired.