This pudding will keep for 3 days in the fridge. To reheat, put in the microwave for 1 minute.
Preheat oven to 350°F. Grease a shallow 11-by-7-inch (2-quart) baking dish. In a medium bowl, stir brown rice, milk, cinnamon stick, 1/4 cup sugar, vanilla and salt until combined. Pour mixture into baking dish.
Spread remaining 2 Tbsp. sugar on a flat plate. Dip bananas in sugar and nestle into pudding mixture. Bake until milk is almost completely absorbed and bananas are tender, 25 to 30 minutes. Let stand for 5 minutes, then discard cinnamon stick and serve.