In a saucepan, heat the milk, sugar, and cinnamon stick over medium heat until it starts to simmer. Do not let it boil.
In a separate bowl, whisk the egg yolks until they are pale and slightly thickened.
Slowly pour a ladle of the hot milk mixture into the egg yolks, whisking constantly to temper the eggs.
Pour the egg mixture back into the saucepan with the remaining milk, whisking constantly.
Cook over low heat, stirring constantly, until the mixture thickens enough to coat the back of a spoon.
Remove from heat and strain the mixture through a fine mesh strainer to remove any cooked egg bits.
Stir in the rum (if using) and vanilla extract.
Let it cool, then transfer to a bottle or jar and refrigerate until chilled.
Serve chilled, and enjoy!