Preheat the oven to 350 degrees F. Butter a 10 by 14-inch baking dish with 1 tablespoon of the butter and set aside.

Melt the remaining 8 tablespoons butter in a large skillet over medium heat. Add 1 cup of the brown sugar and the cinnamon and cook, stirring, until the sugar dissolves, about 2 minutes. Add the bananas and cook on both sides, turning, until the bananas start to soften and brown, about 3 minutes. Add the banana liqueur and stir to blend. Carefully add the rum and shake the pan back and forth to warm the rum and flame the pan. (Or, off the heat, carefully ignite the rum with a match and return to the heat.) Shake the pan back and forth, basting the bananas, until the flame dies. Remove from the heat and let cool.

Whisk together the eggs, remaining 1/2 cup brown sugar, the cream, milk, and vanilla in a large bowl. Add the cooled banana mixture and bread and stir to blend thoroughly. Pour into the prepared baking dish and bake until firm, 50 minutes to 1 hour. Cool on a wire rack for 20 minutes.

To serve, scoop the pudding onto dessert plates. Top each serving with a small scoop of vanilla ice cream, drizzle with caramel sauce, and serve immediately.