Cookie Crust

1 cup finely ground chocolate cookie crumbs (such as Nabisco Famous Chocolate Wafers; about 20 cookies)
2 tablespoons unsalted butter, melted
1/8 teaspoon kosher salt

Budino

3 cups whole milk, divided
1/4 cup cornstarch
3/4 cup (packed) dark brown sugar
5 large egg yolks
3 tablespoons unsalted butter
2 teaspoons dark rum
1 teaspoon kosher salt

Salted Caramel Sauce

3/4 cup plus 2 tablespoons heavy cream, divided
1/2 vanilla bean, split lengthwise
1/2 cup sugar
2 tablespoons light corn syrup
4 tablespoons (1/2 stick) chilled unsalted butter, cut into 1/2" cubes
1/4 teaspoon kosher salt
Lightly sweetened whipped cream