In a medium saucepan, heat the sugar over medium-high heat, stirring frequently with a rubber spatula as it melts. When it has melted completely and drips smoothly off the spatula (with no discernable sugar granules), remove the spatula and continue to cook the sugar without stirring. Watch the pan closely—the caramel will turn darker in color. Watch for smoke to rise off the top. When the caramel starts to smoke, count to ten and remove the pan from the heat.
Using an oven mitt, carefully add the butter to the pan. It might spatter when it hits the hot caramel, so be careful. Stir the mixture with a rubber spatula until the butter has melted and the mixture is smooth. Stir in the salt. Slowly pour in the cream—it will bubble up, but you must continue to stir it as you pour, so the cream incorporates smoothly into the caramel. Be careful!
Remove the pan from the heat; add the vanilla and stir to combine.
Let cool a little before serving, or store the caramel sauce in an airtight container in the refrigerator for up to 2 weeks. (The butter might start to separate, but if you reheat it, you can whisk it until the mixture is combined again.)