Grease 13x9 inch oblong pan with butter; set aside. Adjust oven rack to middle position and preheat oven to 350°F. Place bread cubes in large bowl.
In medium saucepan whisk milk and sweetened condensed milk combine. Heat over medium heat, stirring occasionally, until scalding. Remove from heat and whisk in chocolate until melted and smooth.
In small bowl, whisk vanilla and eggs to combine. Slowly whisk 1/2 cup warm milk mixture into egg mixture to combine. Whisk this mixture back into milk mixture. Pour 1/2 mixture over bread cubes and let soak into bread, stir gently. Pour into pan. Sprinkle pecans over top.
Combine cream and sugar for caramel sauce. Heat milk and sugar until sugar melts. Stir in remaining ingredients. Drizzle over pudding. Pour remaining milk mixture over caramel.
Cover with foil and bake approximately 1 hour. Remove foil and continue to bake until slightly brown, about 15 min.
While pudding cools slightly, make rum sauce.
Combine powered sugar and rum. Drizzle over warm pudding. Serve warm.