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