For the budino: In a medium bowl, whisk together the egg yolks, the 2 tablespoons cream, the 2 tablespoons milk and the cornstarch.
In a medium saucepan, combine the remaining 1 1/2 cups cream, the remaining 1 1/2 cups milk and the kosher salt. Open the vanilla bean and, with the tip of the paring knife, scrape the seeds into the saucepan. Add the pod as well and bring the mixture to a simmer over medium heat.
In another medium saucepan heat the granulated sugar over medium-high heat, stirring occasionally with a wooden spoon or heatproof spatula, until the sugar melts and becomes dark brown. Remove the sugar from the heat and slowly stir in the warm cream mixture (be careful, as it will bubble and sputter) until well incorporated.
Whisk a small ladleful of the warm cream mixture into the egg yolk mixture a little at a time. Then slowly mix the egg yolk mixture into the saucepan. Cook, stirring, until a thick custard forms.
Strain through a fine-mesh strainer into a large heatproof measuring cup (you will have about 3 3/4 cups) or a bowl. Whisk in the butter until melted.
While the custard is still warm, divide it among the serving bowls or cups (we use plain juice glasses), filling them no more than three-quarters full. Cover with plastic wrap, pressing it onto the surface to prevent a skin from forming, and refrigerate for at least 2 hours and up to 24 hours.
For the salted caramel sauce: In a small saucepan, combine the cream and salt. Open the vanilla bean and scrape the seeds into the cream with the tip of a paring knife. Add the pod to the saucepan as well, and bring the cream to a simmer over medium heat.
Meanwhile, combine the butter and the sugar in a medium saucepan, and cook, stirring occasionally with a wooden spoon or heatproof spatula, until the butter and sugar have melted and are a deep amber color. (The caramel will bubble and look strange as it cooks. Once the sugar melts and the moisture from the butter evaporates, the mixture should become thin and glossy.) Remove the sugar from the heat and carefully add the hot cream; it will bubble and spurt but will settle down. Stir well to combine, then strain though a fine-mesh strainer. The sauce will keep, covered and refrigerated for at least 1 week. Reheat it gently before using.
To serve: In a medium bowl, combine the heavy cream, confectioners' sugar and vanilla extract and whisk to medium peaks. Pour 2 to 3 tablespoons of the Salted Caramel Sauce over each custard. Top with a dollop of whipped cream and a pinch of the sea salt.