For the bread pudding: Preheat the oven to 200 degrees F. Slice the bread and bake until dried out, about 20 minutes.
Combine the cream, brown sugar, milk, granulated sugar, rum, vanilla, cinnamon, egg yolks and bananas in a large container, and then blend with a hand mixer.
Raise the oven temperature to 300 degrees F.
Thinly slice the bread and place in an 8-by-10-by-2-inch pan. Pour the pudding into the pan and mush all bread by hand so that the liquid is absorbed and the bread becomes very soggy. Be sure to flatten all of the lumps.
Cover the pan with foil and bake until a skewer inserted in the center of the pudding comes out dry, 2 hours 30 minutes.
Remove the foil and bake until golden, 20 minutes more.
For the sauce: Melt the butter in a saucepot. Add the brown sugar and cinnamon, and mix well. Add the bananas. Add the rum and banana liqueur, and ignite, allowing the flames to subside. Simmer over medium heat until the sauce coats the spoon, about 2 minutes.
Cut the bread pudding into 3 1/2-inch squares. Top each serving with the foster sauce and garnish with mint leaves, then serve.
Use extreme caution when igniting alcohol. Remove the pan from the heat source before adding the alcohol. Pour the alcohol into the pan and carefully ignite with a match or click lighter. Return the pan to the heat and gently swirl to reduce the flames.