Coat an 8-inch glass baking dish with 1 tbsp. butter. Trim the ends off the bread loaf, then slice the loaf crosswise into 10 slices, about 3/4 inch thick. Using 1/2 cup dulce de leche, spread a thin layer on half of the bread slices. Sprinkle with the salt; top with the banana slices. Sprinkle one-third of the chocolate on top, then cover with the remaining bread slices to form sandwiches.
Cut each sandwich into 3 triangles. Arrange the pieces in the prepared dish, placing some flat and others on their crust edge to create a craggy top. Sprinkle the remaining chocolate over the bread.
In a medium bowl, whisk together the eggs and sugar. Whisk in 2 cups milk and the vanilla. Pour the milk mixture over the bread in the dish. Melt the remaining 2 tbsp. butter and drizzle over the top. Let stand for 1 hour or chill for up to 12 hours.
Position a rack in the center of the oven and preheat to 325 degrees . Bake the bread pudding until it puffs up and is springy in the center, 45 to 55 minutes (if the bread pudding was chilled, bake for an extra 5 to 10 minutes). Let cool for 10 minutes.
Meanwhile, in a small saucepan, whisk the remaining 1/2 cup dulce de leche with the remaining 3 tbsp. milk over medium heat until smooth and beginning to bubble, about 3 minutes. Transfer to a small pitcher.
Cut the bread pudding into squares or scoop it with a spoon. Serve in bowls with ice cream and the dulce de leche sauce.