Butter a 10 x 14-inch baking dish with 1 tablespoon of the butter and set aside.
Melt 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. Remove from the heat and let cool.
Whisk together the eggs, remaining 1/2 cup brown sugar, cinnamon, vanilla, salt, milk, and cream in a large bowl. Add the cooled banana mixture, nuts, and bread and fold to blend thoroughly. Pour into the prepared baking dish.
Cover with foil and set aside in refrigerate over night.
Preheat the oven to 325°F Bake for 45 min to 1 hour until firm, remove cover and bake for additional 10 min until top is brown.
Cool on a wire rack for 20 minutes.
For the white chocolate sauce: bring the 1/4 cup sugar, milk, cream, vanilla seeds and empty bean pod almost to a boil in a med sauce pan then remove the bean pod.
In a separate mixing bowl, whisk together the 2 tablespoons sugar, cornstarch and egg yolks in a mixing bowl. Stir until smooth. Gradually add the 3 tablespoons of heavy cream. Add one fourth of the hot milk mixture to the egg yolk mixture. Pour this back into the cream mixture in saucepan.
Cook over a gentle heat, whisking continuously to avoid scorching. Cook until the sauce begins to thicken. Remove from heat and pour the sauce back into the mixing bowl through a fine mesh strainer over white chocolate. Mix until chocolate is melted and sauce is smooth. Cool over an ice bath, stirring occasionally (makes 5 cups).
To serve, scoop the pudding onto dessert plates. Top each serving with a small scoop of ice cream, drizzle with sauce, and serve immediately.