Melt butter in a large skillet over medium heat. Add 1 cup of the brown sugar and the cinnamon. Cook, stirring, until the sugar dissolves, about 2 minutes.
Add the bananas and cook until the bananas start to soften and brown, turning so they cook on both sides, about 3 minutes. Carefully add the rum and shake the pan back and forth to warm the rum and flame the pan. (Or, take the pan off the heat and 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.
Place bread pieces in a large bowl. In a separate bowl, mix together the remaining 1 cup of brown sugar, the eggs, half and half, vanilla and salt. Pour this liquid mixture over the bread pieces and stir until the bread is well coated. Let sit for 10 minutes to allow bread to soak in the liquid, mix with hands to break up bread cubes.
Stir in the cooled banana- rum mixture. Pour the bread mixture into a 9x13-inch baking dish that has been coated with nonstick cooking spray.
Put the filled baking dish inside a large roasting pan. Place the roasting pan in a 350-degree oven and carefully pour hot water into the roasting pan until it comes about 1⁄2 way up the sides of the filled baking dish, being careful not to pour water on the bread pudding mixture. (This ‘water-bath’ method will ensure even cooking). Bake for approximately 90 minutes (lightly cover with foil after 60 minutes if the top begins to brown too much).
Carefully remove the roasting pan from the oven and let cool until you can take the baking dish out of the water. Serve warm or at room temperature with whipped cream, vanilla ice cream or Custard Sauce (recipe follows). Once cooled keep covered in the refrigerator.
1/8 cup sugar
3 egg yolks
1⁄2 cup skim milk
1⁄2 teaspoon vanilla extract
In a medium saucepan mix together the egg yolks and sugar until creamy. Stir in the milk and vanilla. Once all ingredients are combined, cook over low heat until the custard thickens and coats the back of a wooden spoon (approximately 7 minutes). Serve warm or chilled.