Put butter in a medium saucepan and sprinkle with sugars. Cook over medium heat until butter is melted, whisking to combine; then cook, stirring constantly with a heatproof flexible spatula, until butterscotch begins to smoke, 5 to 10 minutes. Reduce heat to medium low and continue cooking and stirring until butterscotch is reddish brown and has the consistency of peanut butter, about 10 minutes.
Remove from heat and very carefully whisk in 2 tbsp. water to stop the cooking. Whisk in cream, milk, and salt. Gently reheat (the acid in the sugars will cause it to break if overheated). Drink right away, or pour into a warm Thermos for sipping later.
Make ahead: Up to 2 days, chilled. Reheat as instructed above.