Preheat oven to 400 degrees F (200 degrees C). Grease a baking sheet with cooking spray; place rolls on top, 2 inches apart. Reserve icing.
Bake in the preheated oven until golden brown, 8 to 10 minutes. Cool on the baking sheet for 5 minutes. Transfer to a wire rack. Let stand, uncovered, until stale, 8 hours to overnight.
Preheat oven to 300 degrees F (150 degrees C). Grease a 2-quart baking dish with cooking spray.
Cut stale cinnamon rolls into 1/2-inch cubes. Whisk heavy cream, brown sugar, 2/3 cup milk, egg yolks, vanilla extract, 3/4 teaspoon cinnamon, and salt in a large bowl. Toss in cinnamon roll cubes until evenly coated. Let stand, tossing occasionally, until cubes being to absorb cream mixture, about 30 minutes.
Pour mixture into the prepared baking dish. Cover with aluminum foil.
Bake in the preheated oven until almost set, about 45 minutes.
Mix 2 tablespoons white sugar and 1/4 teaspoon cinnamon together in a small bowl. Rub in 1/4 cup butter with your fingers until topping is in pea-sized crumbs.
Uncover baking dish and sprinkle topping on top. Continue baking, uncovered, until custard is set, 20 to 25 minutes. Increase oven temperature to 450 degrees F (230 degrees C). Bake until bread pudding forms a golden crust, about 2 minutes more.
Warm reserved icing in a small microwave-safe bowl, about 15 seconds. Drizzle hot icing carefully over bread pudding. Cool on a wire rack, about 30 minutes.
Melt 2 tablespoons butter in a saucepan over medium heat. Mix 1/2 cup sugar and cornstarch together; stir into the melted butter. Pour in milk and bourbon, stirring frequently until mixture begins to boil. Cook, stirring constantly, until thickened, about 5 minutes. Remove from heat.