"It sounds fancy, but budino simply means 'pudding' in Italian. This one is layered with shortbread crumbs and caramel, so if you dig deep, you get a sweet surprise in every bite. If you like, reserve about 2 tablespoons of the crushed cookies to sprinkle on top."
Mix together brown sugar, cornstarch, and salt in a saucepan.
Whisk together egg yolks and milk in a small bowl; gradually whisk into sugar mixture. Cook over medium heat, whisking constantly, until mixture thickens and boils, about 6 minutes. Boil, stirring, for 1 minute. Remove from heat. Stir in butter and 1 teaspoon vanilla extract. Pour into another bowl; chill pudding for 20 minutes.
Divide crushed cookies evenly among 6 half-pint Mason jars or parfait glasses. Layer 1/4 cup pudding, 1 tablespoon caramel sauce, and 1/4 cup additional pudding into each jar. Chill for 30 minutes or up to 1 day.
Beat together cream, white sugar, and remaining 1/2 teaspoon vanilla extract with an electric mixer in a chilled bowl until soft peaks form. Cover and chill.
Just before serving, top each jar generously with whipped cream, drizzle with remaining caramel sauce, and sprinkle with sea salt.