Preheat the oven to 350°F.
Cut the croissants into 1-inch pieces and place them in a large bowl.
Combine the brown sugar, milk, butter, cinnamon stick, vanilla, and dark rum in a medium saucepan over low heat, stirring constantly until the sugar dissolves.
Whisk the eggs in a small bowl and temper them by slowly whisking in the warm milk mixture a few tablespoons at a time, then stir the egg mixture into the remaining custard.
Carefully pour the custard over the croissants and fold the bread until it is thoroughly coated.
Cover the mixture and let it rest for 10 minutes to allow the croissants to fully absorb the custard.
Transfer half of the bread mixture to a greased 9x9-inch casserole dish, sprinkle with half of the chocolate chips, and top with the remaining bread and chocolate chips.
Cover the dish with tented aluminum foil and bake for 30 minutes, then uncover and bake for an additional 15 minutes.
Combine the heavy cream, white sugar, and butter in a medium saucepan over low heat without boiling; once the sugar dissolves, stir in the rum and pour the warm sauce over the bread pudding slices.
