Combine the brown sugar, water, and salt in a heavy bottomed saucepan, and place the pan over medium high heat. Cook until the caramel is very dark, and almost smoking, which should take about 10 minutes.
Standing back with caution, add the cream and milk to the caramel. This will probably make the mixture seize up at first, but if you keep whisking and let it heat back up, it will combine. Bring this mixture back up to a boil, then turn the heat down to medium.
In a large bowl, whisk together the egg, egg yolks, and cornstarch.
Now we’re going to temper the caramel mixture into the eggs by adding a cup of caramel to the eggs, whisking constantly, then pouring the egg mixture back into the pan.
Increase the heat to bring this mixture to a boil, and keep whisking as the custard thickens from the cornstarch, which should take about two minutes.
Remove the pan from the heat and add the butter and rum.
Strain the custard through a fine strainer to remove any big lumps that might have formed, and portion the custard into ramekins.
Cover the ramekins with plastic wrap and chill the custard for at least 3 hours. It's best around 6-8 hours ahead, or even made a day ahead.
To serve, pour on a thin layer of caramel sauce, then add a dollop of whipped cream. Enjoy!