In a medium saucepan set over medium-low to medium heat, melt the butter. Mix in the brown sugar and stir for 2 minutes. Mix in the heavy cream and stir continuously for 2 more minutes (the sauce should be gently bubbling, so adjust the heat, if necessary). Remove the pot from the heat and stir in the vanilla and the salt until well incorporated.
Pour the hot caramel sauce into a jar and allow it to cool completely. Cover and store in the refrigerator for up to a month.
Be sure your saucepan is deep enough to account for the bubbling caramel that may rise up the sides of the pot as it cooks.
The caramel sauce will be thin when it is done cooking, but it will thicken up nicely at room temperature, and even more so once chilled.