Make sure to freeze the container of your ice cream maker at least 24 hours in advance.
Add heavy cream, milk, sugar, and salt to a saucepan and heat over medium. Whisk to dissolve the sugar and salt, making sure the mixture does not start to bubble. Once completely dissolved, remove from heat and let cool briefly.
Pour into a bowl or large (4 cups or larger) liquid measuring cup. Add the eggnog, vanilla, rum (if using), and vanilla, and whisk to combine. Cover and chill mixture in the refrigerator at least 4 hours.
When ready to churn, set up your ice cream maker. Whisk liquid to reincorporate ingredients, then pour into ice cream machine. Churn according to manufacturer's instructions.
Serve directly out of the ice cream maker or store in containers.