Stir in the vanilla extract and rum. Cover and refrigerate for at least 2 hours or overnight.
Pour the custard through a fine-mesh strainer into a clean container. Place the container in an ice bath and stir the custard occasionally until it is cool, about 20 minutes.
Heat the mixture over medium heat, stirring constantly and scraping the bottom as you stir. Cook until the mixture thickens enough to coat the back of a spatula or wooden spoon, 1 to 2 minutes after reaching a bare simmer. Remove the vanilla bean pod.
In a medium bowl, whisk the egg yolks. Slowly whisk the cream mixture into the yolks, then pour everything back into the saucepan.
Heat to a bare simmer over medium heat, stirring frequently. Remove from the heat and cover. Set aside for 30 minutes to allow ice cream to cool slightly and absorb vanilla flavor.
In a medium saucepan, combine the cream, milk, sugar, and salt. Split the vanilla bean and scrape the seeds into the cream mixture, then add the bean pod as well.