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.
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 thermometer reads 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 an instantread 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. DO AHEAD: Can be made 1 day ahead. Keep chilled.
For salted caramel sauce:
Place cream in a small pitcher. Scrape in seeds from vanilla bean; add bean. 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 cold butter and salt. Strain into a heatproof measuring cup. Let cool slightly.
Spoon 2 tablespoons caramel sauce over each budino. Top with whipped cream. Sprinkle some of reserved cookie crumbs over each.
*The original recipe calls for 1/4C cornstarch, but I increased it to ensure a proper set.