Melt the butter in a small saucepan over medium heat.
Stir in the brown sugar and continue to heat until the sugar has dissolved.
Slowly stir in the heavy cream and bring the mixture to a gentle bubble - holding it at a bubble for 5 minutes. Continue to stir frequently throughout this time.
Stir in the rum, vanilla, and salt and cook for 1-2 more minutes to thicken.
Remove and let it cool for 5-10 minutes before serving. Serve warm.