01 - Preheat oven to 350°F (175°C). In a large bowl, vigorously whisk together milk, eggs, 1 cup brown sugar, and cinnamon until fully blended.
02 - Add cubed bread to custard, ensuring all pieces are well coated. Allow bread to absorb mixture for 10 minutes, then fold in sliced bananas.
03 - Transfer mixture to a greased 9x13-inch baking dish and smooth the top. Bake uncovered for 45 minutes, or until golden and set in the center.
04 - While pudding bakes, melt butter in a skillet over medium heat. Add 1/2 cup brown sugar and stir until dissolved. Pour in rum and simmer for 2 minutes, stirring constantly until sauce thickens.
05 - Spoon warm bread pudding onto plates and drizzle generously with rum sauce before serving.