Melt the butter in a medium heavy-bottomed saucepan (not non-stick) over medium heat.
Add the brown sugar, salt and heavy cream and stir the mixture with a rubber spatula until it's well blended. Bring the mixture to a boil, scraping down the sides occasionally, and boil it for 4 to 5 minutes.
Remove the mixture from the heat and stir in the vanilla extract. Transfer the butterscotch sauce to a container to cool. (See Notes.)
The butterscotch thickens significantly as it cools, so it's important to not over-boil it or you'll end up with chewy butterscotch caramels rather than a pourable sauce.
If you want to add a kick to your butterscotch sauce, stir in 1 to 2 tablespoons of your choice liquor (think bourbon, whiskey or cognac) at the same time you stir in the vanilla extract.