The Mexican chocolate ice cream:
Heat the cream, milk, and cinnamon sticks, and bring to a boil.
Remove the saucepan from the heat, cover, and let infuse for 15 minutes, then remove the cinnamon sticks.

Beat the egg yolks and sugar in a bowl until the mixture is thick and foamy.
Gradually pour in the hot liquid, whisking constantly.
Return the mixture to the saucepan and cook over low heat, stirring constantly.
When the cream coats the spoon, transfer it to a bowl, over the chopped chocolate.
Mix until the chocolate is melted.
Let it cool to room temperature before refrigerating it to make it very cold.

Fold the whipped cream into the cooled custard, then pour the mixture into the ice cream maker and churn according to the manufacturer’s instructions.

Cinnamon swirl:
In a saucepan, bring the brown sugar, corn syrup, and ground cinnamon to a boil for 2 minutes.
Add the cream and mix well.
Off the heat, add the butter, mix, and let cool completely.

Add the cinnamon swirl to the ice cream maker 15 seconds before the end of the churning, so it doesn’t mix too much with the ice cream.