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Pour the 4 cups of milk in a pot on the stove. Add the disk of Mexican chocolate and heat over medium heat until the tablet is melted and incorporated.
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Pour 1 1/2 cups of the hot chocolate into a large mixing bowl and add the 2/3 cup of sugar. Stir until the sugar is dissolved.
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Add the cocoa powder, vanilla, cinnamon, and cayenne pepper and mix well.
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Add the 1 1/2 cups of heavy whipping cream and mix well.
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Put in the refrigerator to chill several hours or overnight.
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When chilled, pour into your ice cream maker and follow the manufacturer's directions.
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When the ice cream reaches soft-serve consistency, transfer to an airtight container and let harden in the freezer for several hours.