Combine milk, sugar, baking soda, and cinnamon in a large pot over medium-high heat. Bring mixture to a boil. Reduce the heat and simmer for 30 minutes.
Whisk egg yolks in a heat-proof bowl. In a slow and steady stream, add about 1 cup of the milk mixture to the eggs, whisking continuously.
Return the milk-yolk mixture to the pot and cook over low heat for 5 to 7 minutes or until the consistency is that of half and half.
Remove from heat immediately and transfer contents to a bowl. Keep bowl chilled over an ice bath.
Discard the cinnamon and whisk in rum and vanilla extract.
Chill completely before serving.