Preheat oven to 350℉. Place a large skillet over medium heat. Add the 4 tablespoons of butter. Once the butter melts, add the sliced bananas. Saute the banans to soften them, about 5 minutes, then add the brown and granulated sugars. Reduce heat to low and stir until sugars are dissolved, about 2 minutes. Remove from heat and stir in the vanilla, rum, and cinnamon. Let cool for 10 minutes.

Butter the sides of an 8x10’’ casserole dish (it needs to hold 10-12 cups). Set aside.

In large bowl, combine the bread cubes, milk, and half and half. Use a rubber spatula to press the bread into the liquid.

Whisk a little bit of the warm banana mixture into the eggs. Add this egg mixture to the full skillet of banana mixture and stir to combine. Add this to the bowl with the bread and milk mixture. Fold together with a rubber spatula and then spread evenly into the prepared pan.

Bake the pudding for 40-50 minutes until it has set and barely jiggles when you shake the pan. It will be golden brown on top, slightly crispy on the outside but moist and tender on the inside. Serve hot with vanilla butter sauce spooned over the top.

In a medium saucepan, combine the butter, cream, and both sugars. Melt together over medium heat for 5-8 minutes, stirring occasionally, until the mixture thickens slightly and comes to a boil. Boil for 1 minute, stirring continuously, then remove from heat. Stir in the vanilla and rum, then place in a gravy boat to serve immediately over hot bread pudding.