4 large eggs
2 cups whole milk
¼ cup plus ½ cup brown sugar
2 teaspoons vanilla extract
1 tablespoon plus ½ cup dark rum
1 teaspoon plus ½ teaspoon kosher salt
6 cups cubed brioche, approximately ½ loaf
5 ripe bananas, peeled and cut into coins
1 tablespoon white sugar
5 tablespoons unsalted butter
¼ cup heavy cream
Preheat oven to 350. In a large bowl, whisk together the eggs, milk, ¼ cup of brown sugar, vanilla, 1 tablespoon rum and salt. Add the bread, and stir so that it absorbs the liquid. Set aside to soak.
Butter a 2-quart baking dish, and place it on a baking sheet. Line the bottom of the dish with banana coins, then spoon about ½ of the soaked bread over them. Top with banana coins, and spoon the remaining soaked bread over the top. Bake for 40 minutes, then sprinkle with the tablespoon of white sugar. Continue baking until the pudding is puffed and golden on top, approximately 50 to 60 minutes total.
Meanwhile, make the sauce. Melt the butter in a small sauce set over medium heat. Add the remaining brown sugar, cream, remaining rum and salt, and whisk until combined. Simmer, whisking, until the sauce has thickened, approximately 5 to 8 minutes. Remove from heat and keep warm.
Serve the pudding warm, with some of the sauce drizzled over the top, alongside vanilla ice cream.