Cookie Crust:
1 cup chocolate cookie crumbs, finely ground (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 dark brown sugar, packed
5 large egg yolks
3 tablespoons unsalted butter
2 teaspoons dark rum
1 teaspoon kosher salt
Cheesecloth, for straining

Salted Caramel Sauce:
3/4 cup heavy cream
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-inch cubes
1/4 teaspoon kosher salt
Lightly sweetened whipped cream

For the Cookie Crust:
Mix cookie crumbs, butter, and salt in a medium bowl. Place 2 tablespoons in a small bowl; cover and chill for garnish. Press 2 tablespoons cookie mixture onto the bottom of each of eight 8-ounce mason jars or ramekins.

For the Budino:
Whisk 1/2 cup milk and cornstarch in a small bowl; set aside. Heat remaining 2 1/2 cups milk in a small saucepan just to a simmer; set aside. To make caramel, stir sugar and 3/4 cup water in a heavy saucepan over medium heat until sugar dissolves. Increase heat; cook without stirring until an instant-read thermometer registers 210°F-220°F. Line a sieve with a double layer of cheesecloth and set over a large pitcher. Whisk egg yolks in a large bowl. Gradually whisk in hot milk, then cornstarch mixture. Slowly whisk in caramel. Return mixture to saucepan. Whisk constantly over medium heat until mixture thickens and a thermometer registers 175°F, about 3 minutes. Remove from heat; whisk in butter, rum, and salt. Pour through prepared sieve. Pour 1/2 cup budino over crust in each jar. Cover; chill until set, 4-5 hours. Can be made 1 day ahead. Keep chilled. 

For the Salted Caramel Sauce:
Place cream in a small pitcher. Scrape in seeds from vanilla bean; add bean but remember to remove the bean before using the cream. Set aside. Stir sugar, corn syrup, and 2 tablespoons water in a heavy saucepan over medium heat until sugar dissolves. Increase heat to medium high; boil, occasionally swirling pan and brushing down sides with a wet pastry brush, until deep amber color forms, 5-6 minutes. Remove from heat; gradually add vanilla cream (mixture will bubble vigorously). Whisk over medium heat until smooth and thick, about 2 minutes. Remove from heat; whisk in butter and salt. Strain into a heat-proof bowl. Let cool slightly. Spoon 2 tablespoons caramel sauce over each budino. Top with whipped cream. Sprinkle some of reserved cookie crumbs over each.