Step 1

Preheat oven to 350°F. Place bread in large bowl. Drizzle with 3 tablespoons melted butter; toss. Transfer to rimmed baking sheet. Bake until bread begins to color, stirring occasionally, about 12 minutes. Cool. Brush 11x7x2-inch glass baking dish with 1 tablespoon melted butter.

Step 2

Stir whipping cream and 1 cup dulce de leche in medium saucepan over medium heat until blended and bubbling. Remove from heat. Whisk eggs and yolks in large bowl. Add rum, vanilla, and salt; gradually whisk in warm dulce de leche mixture. Stir in bread cubes. Let soak 30 minutes, stirring occasionally.

Step 3

Mix chocolate chips into custard mixture. Pour into prepared dish. Sprinkle with 2 tablespoons sugar. Bake pudding until puffed and set in center, about 35 minutes. Let cool 15 minutes. Dust with powdered sugar. Serve warm, passing more dulce de leche topping alongside.