Step 1

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.

Step 2

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°–220°

Step 3

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°, about 3 minutes. Remove from heat; whisk in butter, rum, and salt. Pour through prepared sieve.

Step 4

Pour 1/2 cup budino over crust in each jar. Cover; chill until set, 4–5 hours. DO AHEAD Can be made 1 day ahead. Keep chilled.

Step 5

Place cream in a small pitcher. Scrape in seeds from vanilla bean; add bean. Set aside.

Step 6

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 heatproof bowl. Let cool slightly.

Step 7

Spoon 2 tablespoons caramel sauce over each budino. Top with whipped cream. Sprinkle some of reserved cookie crumbs over each.