In a large, heavy pot, combine all of the ingredients and bring to a boil, stirring with a wooden spoon until the sugar is dissolved. Reduce the heat and simmer over moderate heat, stirring frequently, until the mixture is reduced to 2 cups and is the consistency of warm caramel, about 40 minutes. Discard the cinnamon stick and let the dulce de leche cool to room temperature.
Transfer the dulce de leche to a blender and puree until smooth. Skim off the foam and transfer to a pitcher.
The dulce de leche can be refrigerated for up to 3 days.