Put the butter in a medium saucepan and add both sugars. Cook over medium heat until the butter is melted, whisking to combine, then cook, stirring constantly with a heatproof spatula or a wooden spoon, until the butterscotch begins to smoke, 5 to 10 minutes. Reduce the heat to medium-low and continue cooking and stirring until the butterscotch is very dark and registers 300°F (150°C)—or very slightly above—on an instant-read thermometer, about 10 minutes.
Remove from the heat and very carefully whisk in the water to stop the cooking. Whisk in the cream, then whisk in the milk, rum, if using, and salt. Gently reheat over low heat, stirring (the acid in the sugars will cause the mixture to break if overheated). Drink right away, or pour into a Thermos for sipping later.
Do Ahead: The butterscotch drink can be made ahead, cooled, transferred to a lidded container, and refrigerated for up to 5 days. Reheat as instructed in step 2 above.