Place the chocolate in a large mixing bowl; set aside.
Pour the heavy cream, milk, brown sugar, and cocoa powder into a medium saucepan. Cook over medium-high heat until the sugar is dissolved and the mixture begins to boil. Immediately pour hot cream mixture over the chocolate and allow to sit for 5 minutes. Stir until smooth.
In a separate bowl, whisk the egg yolks and egg together. Slowly whisk in a few tablespoons of the chocolate cream mixture. Pour the egg mixture into the chocolate cream, and whisk until smooth. Stir in the salt and vanilla. Gently fold in the bread and 2/3 cup of the milk chocolate chips.
Pour mixture into a greased 8-inch square baking dish, gently pressing down on the bread to ensure all of the pieces are saturated in the custard. Allow to sit and soak for at least an hour**.
Preheat the oven to 350ºF. Bake the bread pudding for 35-40 minutes. I baked it covered for the first 20 minutes, then removed the foil for the remaining time to allow it to get slightly crisp on the top. After it's done baking, immediately sprinkle the remaining 1/3 cup of the milk chocolate chips evenly over the bread pudding.