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, apple cider 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 heavy cream.
Use an electric mixer to beat until just thickened but not completely whipped (about 2–3 minutes).
Divide brown sugar mixture among four tall glasses (about ¼ cup for each glass).
Add ¼ 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.
