For pudding: Melt the butter in a medium-size heavy-duty saucepan. Mix in the brown sugar and cook until the mixture bubbles vigorously. Remove from the heat and stir in the heavy cream, adding it 1 tablespoon at a time. Stir in the milk and salt.
In a small bowl, whisk together the cornstarch and eggs until the cornstarch is dissolved.
Cook over medium heat, stirring constantly with a whisk, until the pudding comes to a boil. Reduce the heat to a simmer, continue to stir, and cook for 2 minutes.
Pour the pudding into a container and stir in the bourbon and vanilla. Cover and refrigerate until thoroughly chilled.
To caramelize the bananas: Melt the butter in a large skillet. Stir in the brown sugar, add the banana slices, and cook over medium-high heat, without stirring, until the bananas begin to sizzle.
Add the coffee and vanilla and continue to cook until the juices thicken to the consistency of maple syrup, Avoid stirring the bananas. Instead, tilt the pan and spoon the sauce over the bananas as they cook.
To Serve: Dish up the pudding in individual custard cups or in tall glasses with the banana slices heaped over it.