For the Budino:

1. Combine the brown sugar, water, and salt in a large, heavy-bottomed pot over medium-high heat. Cook to a smoking, dark caramel, about 10 to 12 minutes (the sugar will smell caramelized and nutty and turn a deep brown.) Immediately whisk the cream and milk into the caramel to stop the cooking. The mixture will steam and the caramel will seize, but will become smooth again as you whisk. Bring the mixture to a boil and reduce the heat to medium.
2. In a medium bowl, whisk together the egg, egg yolks, and cornstarch. Temper the hot caramel cream into the egg mixture by adding a cup of caramel at a time, whisking constantly. When half the caramel is incorporated, pour the egg mixture back into the remaining caramel, and boil while whisking constantly until the custard is very thick, about 2 minutes. Remove the custard from the heat and whisk in the butter and rum. Strain the custard through a fine mesh strainer to remove any lumps and divide among ten 6-ounce ramekins. Cover with plastic wrap (gently lay the wrap over the surface of the pudding – this will ensure it doesn’t develop a skin) and chill for several hours, or up to 3 days.

For the caramel sauce:

1. Beat the cream and vanilla (pod and seeds) in a medium saucepan over medium heat until simmering. Add the butter, turn off the heat, and set aside. In a large heavy-bottomed saucepan, combine the sugar, corn syrup, and water. 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. Remove the caramel from heat and carefully whisk the cream mixture (remove the pod first) into the caramel. Be very careful—the caramel will steam and bubble. Whisk to combine. Place the pan in a large bowl of ice water to cool. You can store this in the fridge if you want to make this ahead. When you’re ready to serve, simply microwave it for 20 seconds or so to warm it up.

To serve:

1. Whip the whipping cream in a chilled medium bowl until it begins to thicken. Add the crème fraiche and beat until thick and fluffy. Spoon 1 tablespoon (or 2 tablespoons if you’re making 6 larger portions) of the warm caramel sauce onto each budino, sprinkle with a little of the fleur de sel, and add a dollop of the whipped cream/creme fraiche mixture.