In a blender, combine the fresh or reconstituted masa with the chocolate, cinnamon, sugar and milk. Blend until smooth. Strain into a medium (4-quart) saucepan, set over medium heat and whisk constantly until the mixture comes to a boil and thickens. Remove from the heat, pour into a bowl, cover and refrigerate until thoroughly cold, 2 hours or longer. (Alternatively, you can stir the mixture over an ice bath to cool it down quickly.)
Stir the cream into the chocolate base, pour into the bowl of your ice cream maker and churn according to the manufacturer’s directions. When the ice cream has thickened to the texture of soft-serve, usually about 20 minutes, scrape it into a freezer container, cover and freeze until firm enough to scoop, usually 3 or 4 hours.