Making the Vanilla Bean Pudding : In a saucepan, combine 2 cups of milk, heavy cream, and half the sugar. Split the vanilla bean, scrape out the seeds, and add both seeds and pod to the saucepan. Heat the mixture over medium heat until warm, but not boiling. In a separate bowl, whisk egg yolks, remaining sugar, cornstarch, salt, and remaining ½ cup milk. Slowly pour the warm milk mixture into the egg mixture while whisking constantly to temper the eggs. Return everything to the saucepan and cook over medium heat, stirring constantly, until thickened. Remove from heat, discard the vanilla bean pod, and whisk in butter until smooth. Pour pudding into serving cups or bowls, and chill for at least 2 hours for best texture.

Making the Caramel Sauce : In a heavy-bottomed saucepan, heat sugar over medium heat, stirring continuously until melted and amber in color. Add butter carefully and whisk until incorporated. Slowly pour in heavy cream while whisking; mixture will bubble vigorously. Add a pinch of salt and stir until smooth. Allow to cool slightly before drizzling over the chilled pudding.