In a medium saucepan at low to medium heat, stir together milk, egg yolks and sugar. Add the cinnamon stick.
Cook and stir until the mixture coats the back of a spoon. Do not bring mixture to a boil.
Remove saucepan from the heat and let it cool to room temperature. Remove cinnamon stick.
Stir in heavy cream, rum, and vanilla. Cover and chill for a minimum of 4 hours.
Top each serving with freshly grated nutmeg.