Position an oven rack about 6 inches from the broiler and heat the broiler on high.
In a large bowl, whisk the milk, brown sugar, eggs, vanilla, and salt.
Melt 1 Tbs. of the butter in a 9- to 10-inch cast-iron skillet over medium heat. Add the bread and cook, tossing often, until toasted and golden-brown in spots, about 5 minutes. Add the bread to the milk mixture, toss to coat, and let soak for at least 5 minutes.
Wipe the skillet clean and return to medium heat. Melt the remaining 1 Tbs. butter in the skillet until just browned and fragrant, about 2 minutes. Spread the bread mixture evenly in the skillet. Nestle the chocolate pieces into the bread. Cook until the pudding is browned around the edges and just set in the center, 4 to 5 minutes.
Transfer the skillet to the oven and broil until golden-brown on top and completely set, 1 to 2 minutes more. (Watch closely to ensure that any chocolate pieces poking out of the pudding don’t burn.) Scoop the bread pudding into bowls and serve warm or at room temperature.
nutrition information (per serving):
Photo: Scott Phillips