1.Combine 3 cups heavy cream and milk into a large measuring cup and set the mixture next to the stove. In a large, heavy-bottomed pot over medium-high heat, stir the brown sugar, ½ cup water, and the 1 1/2 teaspoon salt to combine. Cook until the sugar is melted, dark brown, and smells caramelized, 10 to 12 minutes.
2.Immediately whisk in the cream and milk mixture (the mixture will steam and the butterscotch will seize, but will become smooth again as you continue whisking). Bring the mixture to a boil, then lower the heat to medium.
3.In a medium bowl, whisk the egg, egg yolks, and cornstarch. Add 1 cup of the prepared butterscotch, whisking constantly. Whisk in the remaining butterscotch, then boil, whisking constantly, until the custard is very thick, about 2 minutes.
4.Remove from the heat and whisk in 5 tablespoons butter and rum. Strain the custard through a fine mesh strainer to remove any lumps, and divide among ten 6-ounce ramekins or serving bowls. Cover with plastic wrap and refrigerate until completely chilled, several hours or up to 3 days.
5.In a medium saucepan over medium heat, add the 3/4 cup cream and vanilla extract and bring to a simmer. Add 2 tablespoons butter, turn off the heat, and set aside.
6.In a large, heavy-bottomed saucepan, combine the sugar, corn syrup, and 2 tablespoons 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 vanilla cream into the caramel (stand back as the mixture will steam and bubble vigorously); continue whisking until completely combined. Let cool significantly (mixture should be just warm for serving).
7.In a chilled bowl of an electric mixer with whisk attachment, whip the crème fraiche on high speed just until thickened (15–30 seconds). Top each budino with 1 tablespoon of the warm caramel sauce, a dollop of the whipped crème fraîche, and a pinch of sea salt.