In a medium saucepan, combine the brown sugar and balsamic vinegar, bring to a rolling boil, stirring occasionally. Once the mixture is boiling rapidly, cook for an additional 3-4 minutes.
Add the heavy cream. The mixture will hiss and spit (be careful), then start to assimilate with the balsamic caramel. Whisk together until smooth and remove from heat. Stir in the butter and salt.
Transfer the balsamic caramel dessert sauce to a glass jar. Cool to room temperature, then cover and refrigerate.
Spoon balsamic caramel sauce over strawberries or ice cream (or both).