Stir the brown sugar and water together in a small saucepan over medium heat; bring to a gentle boil, and cook until the mixture reads 240 degrees F (115 degrees C) on a candy thermometer. When a small amount of the syrup is dropped into a glass of cold water, it should form a soft ball. Add the butter, salt, cider vinegar, and 1/4 cup heavy cream, and stir. Remove from heat and set aside to cool completely; mix the rum extract into the cooled mixture.
Combine 2 tablespoons of the brown sugar-cream syrup with remaining 1/2 cup heavy cream in a non-plastic bowl; beat with an electric mixer until whipped and cream holds soft peaks, 2 to 3 minutes.
Pour about 1/4 cup of the remaining brown sugar-cream mixture into a tall glass; stir about 1/4 of a bottle of cream soda into the cream mixture. Top with the remainder of the cream soda in the bottle. Top with about 1/4 of the whipped cream. Repeat with 3 more glasses to serve.