For the gelato: Add the milk, cream, 1 cup of the sugar and vanilla bean to a large heavy-bottomed saucepan and bring to a simmer over medium-high heat. Stir occasionally to combine.
Meanwhile, whisk together the egg yolks with the remaining 1/4 cup sugar in a large heatproof bowl until they become lighter in color and the sugar dissolves.
When the milk mixture is at a simmer, lower the heat to medium. Carefully ladle 1 cup of the hot milk mixture into the egg mixture, while whisking, and whisk until fully combined. Repeat this step one more time, then pour the egg mixture back into the saucepan. Add the salt. Cook the mixture while stirring constantly with a wooden spoon, making sure to stir the corners of the pot. Cook until the mixture thickens and can coat the back of a spoon.
Prepare an ice bath in a large bowl or pot with water and ice. Carefully strain the mixture into a strainer set over a heatproof bowl. Remove and discard the vanilla bean. Put the heatproof bowl in the ice bath. Stir occasionally until the mixture is chilled and no longer warm. Set in the fridge until very cold, at least 1 hour.
Freeze the mixture to an ice-cream maker and process according to the manufacturer's instructions. Transfer to a freezer-proof container and cover.
Freeze before serving, 1 to 2 hours. It's best to serve the gelato the same day it's made.
For the compote: Combine the cherries, 1/2 cup water and the sugar in a saucepan and bring to a bare simmer over medium heat. Stir to allow the sugar to melt. Cook the compote until the ingredients combine and the cherries just begin to break up, 7 to 10 minutes. (Depending on the ripeness of your cherries you may want to cook them longer.) Remove the pan from the heat and add the rum. Ignite the compote safely with a long-handled lighter and return to the stovetop to flambe and cook off the alcohol.
Serve the gelato with the warm compote.