Combine milk, sugar, and cinnamon sticks in a large saucepan and bring to a boil over low heat. Continue to boil mixture, stirring constantly, until milk has reduced by a little more than 1/3, about 20 minutes. Remove from heat and allow to cool.
Beat egg yolks with an electric mixer until thick and pale. Mix in a little of the warm milk. Pour egg yolk mixture into the saucepan and bring to a boil over low heat. Stir constantly and scrape bottom and sides of saucepan until mixture is thick enough to coat the back of a spoon, 5 to 7 minutes. Remove from heat, fish out cinnamon sticks, and cool to lukewarm. Stir in rum and cool completely, about 2 hours.
You can also use brandy instead of rum.
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