In a small saucepan over medium heat, combine brown sugar and water.
Bring to gentle boil and cook, stirring often, until the mixture reads 240°F on a candy thermometer.
Stir in butter, vinegar, and a third of the heavy cream.
Set aside to cool to room temperature. Once the mixture is cooled, stir in the rum extract.
In a medium bowl, combine 2 tablespoons of brown sugar mixture and the remaining 1/2 cup of heavy cream. Use an electric mixer to beat until just thickened, but not completely whipped (about 2–3 minutes).
Divide the brown sugar mixture among four tall glasses (about 1/4 cup for each glass). Add 1/4 cup of cream soda to each glass. Stir to combine. Fill each glass near to the top with additional cream soda and add the whipped topping.