Combining: Mix the condensed and evaporated milk and the egg yolks. Stir until the mixture is well combined.Sieve, then discard solids. Add cinnamon sticks and nutmeg.
Cooking: Boil in a large pot in a bain-marie (hot water bath) over medium-low heat for 20 minutes, lower heat and simmer for 20 more minutes. Stir often to prevent it from sticking to the bottom.
Cooling: Add the rum and remove it from the heat. Stir constantly but slowly until it cools down to room temperature.Remove the cinnamon sticks.
Serving: Chill, then ladle into a bottle for storage, or to serve immediately.
Video
Notes
Don't worry if it's not as thick as you're used to when you finish cooking, it will get thicker when chilled.