3 cups heavy cream (I ended up using slightly less cream and more milk, due to availability, and it was also fine)
1 1/2 cups milk
1 large egg
3 large egg yolks
5 tablespoons cornstarch
1 1/8 cups dark brown sugar
1 1/2 teaspoons coarse salt
5 tablespoons unsalted butter, cut into chunks
1 1/2 tablespoons dark rum (or brandy or whiskey)
1/2 cup heavy cream
Scrapings from 1-inch piece of vanilla bean, or 1/4 teaspoon vanilla extract
2 tablespoons butter
2 tablespoons light corn syrup
1/2 cup sugar
1 1/4 teaspoons fleur de sel or other flaky salt, for topping
1/4 cup cream
3/4 cup crème fraîche or sour cream (I just used straight whipped cream, and it was totally delicious)
To make the budino: Measure out your milk and cream into a bowl/pitcher/measuring cup, and set aside. In a separate medium bowl, whisk together the egg, egg yolks, and cornstarch, and set aside. Have ten little pudding dishes and a strainer at the ready.
Place the brown sugar, coarse salt, and 1/2 cup water in a pot. Heat over medium-high, tipping the pot to swirl occasionally, until the mixture foams and thickens and caramelizes to a dark, nutty brown, ~10 minutes (since it's brown sugar, it's harder to tell the color change than with ordinary sugar, but you want it to darken significantly).
When the mixture darkens, you need to stop it before it burns, so immediately dump in the cream and milk. Stand back — it'll sputter and seize! Cook the mixture until the hardened sugar melts, and can be smoothly whisked together with the milk, and the entire pot is steaming on the edge of a simmer.
Gradually pour the hot caramel and milk mixture into your egg mixture. Pour in slowly, whisking, until it is incorporated into a smooth mixture (you'll need to go bit by bit at first, then you can add larger amounts). Pour back into your pot, then return to a medium heat and whisk constantly until the mixture becomes very thick, ~2 minutes. Add the butter and rum, let sit for a few moments to heat and soften, then whisk to combine. Pour the mixture through the strainer into your pudding cups, and refrigerate to chill and set, ~3 hours or longer.
To make the caramel sauce: Place the cream and vanilla seeds in a medium saucepan (you can toss in the pod to infuse as well), and heat to simmer. Add the butter, then remove from heat and set aside.
In a large, heavy saucepan, combine the corn syrup, sugar, and enough water (3-4 tablespoons) to make a wet, slightly sandy mixture. Cook over medium-high heat, swirling the pan occasionally so that it mixes evenly, until the mixture is a deep amber, ~10 minutes (I tend to aim for just this side of burnt, but since you're adding a warm rather than cold mixture to this one, it won't cool off quite as immediately so be vigilant). When it's gotten to the desired darkness, remove from heat and carefully whisk in the cream mixture. Set aside to cool. If making in advance, you can just refrigerate, then rewarm before serving.
To make the whipped cream: Whip the cream to soft peaks (under-mixing is better than over-mixing!), and whisk in the crème fraîche or sour cream (or just use a scant cup of cream instead). No sugar, no vanilla. Just whipped dairy.
To assemble: If your caramel sauce has been made in advance, heat gently in a saucepan or microwave until it's warm and pourable. Pour a tablespoon or so of warm caramel over each budino, then sprinkle with a pinch of flaky salt. Add a dollop of your whipped cream, and serve.