Step 1 In a large bowl, combine the cream and milk and set aside.
Step 2 In a large heavy-bottomed pot, combine the brown sugar, one-half cup water and salt over medium-high heat. Cook to a smoking, dark caramel, about 10 to 12 minutes. Sugar will smell caramelized and nutty and turn a deep brown.
Step 3Immediately whisk the cream mixture carefully into the caramel to stop the cooking (the mixture will steam and the sugar will seize). Bring the mixture to a boil and reduce the heat to medium.
Step 4In a medium bowl, whisk together the egg, egg yolks and cornstarch. Temper the hot caramel cream into the egg mixture by adding a cupful of caramel at a time, whisking constantly, until half is incorporated. Pour the egg mixture back into remaining caramel, stirring constantly with a whisk until the custard is very thick and the corn starch is cooked out, about 2 minutes.
Step 5Remove the custard from the heat and whisk in the butter and rum.
Step 6Pass the custard through a fine mesh strainer to remove any lumps and divide among 10 (6-ounce) ramekins leaving one-half inch at the top. Cover with plastic wrap and chill several hours or up to three days.