In a small saucepan, combine the brown sugar and water. Heat, stirring frequently, over medium heat until a candy thermometer registers 240 degrees. Reduce heat to low and, whisking constantly, add in butter, salt, and vinegar and whisk until smooth. Slowly stir in 1/4 cup of the heavy cream, reserving the rest for later, and whisk until smooth and combined and then remove from heat. Add the rum extract and allow to cool to room temperature. You can use this as a sauce/ice cream topping/syrup, or you can save it for the butterbeer.

To make the butterbeer, combine 2 tablespoons of the salted butterscotch with the remaining 1/2 cup of heavy whipping cream and whip with electric beaters until soft peaks form. Set aside.

Pour 1/4 cup of the butterscotch (approximately) into each of the chilled mugs or glasses. whisk in 1/4 cup of cream soda in each glass so it becomes frothy. Then add the remaining cream soda to each glass. Divide the whipped cream mixture evenly among the four glasses and serve immediately.