Step 1In a large bowl, combine the cream and milk and set aside.
Step 2In 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.
Step 1In a medium saucepan, heat the cream and vanilla over medium heat, until simmering, about three minutes. Add the butter, turn off the heat and set aside.
Step 2In a large heavy-bottomed saucepan, combine the corn syrup and sugar. Add enough water to make a wet sandy texture, about one-fourth cup. Cook over medium-high heat, swirling the pan just slightly to gauge the caramelization, until the sugar becomes a medium amber color, about 10 minutes.
Step 3Remove the caramel from heat, carefully whisk the cream mixture into the caramel (be very careful -- it will steam and bubble). Whisk to combine. Place the pan in a large bowl of ice water to cool.
Step 4In a chilled bowl with a wire whisk, beat the whipping cream until it begins to thicken. Add the creme fraiche; whip until thick and fluffy.
Step 5Before serving, warm the sauce over medium heat. Spoon one tablespoon on each budino, sprinkle with one-eighth teaspoon fleur de sel and add a dollop of cream topping.