01 - Preheat oven to 350°F (175°C). Grease a 9x13-inch (23x33 cm) baking dish. In a large skillet over medium heat, melt 2 tablespoons of butter. Add the diced apples, brown sugar, and 1/2 teaspoon of cinnamon. Cook, stirring occasionally, until the apples are softened and caramelized, approximately 6-8 minutes. Remove from heat and set aside.
02 - In a large mixing bowl, vigorously whisk together the eggs, whole milk, heavy cream, root beer, granulated sugar, vanilla extract, 1/2 teaspoon ground cinnamon, and salt until well combined.
03 - Add the cubed bread and the prepared caramelized apples to the custard mixture. Gently toss to ensure all bread pieces are evenly coated. Allow the mixture to soak for 10 minutes to absorb the liquid.
04 - Pour the soaked bread pudding mixture into the prepared baking dish, spreading it evenly. Bake for 40-45 minutes, or until the pudding is golden brown and set in the center.
05 - While the bread pudding is baking, prepare the caramel sauce. In a small saucepan over medium heat, melt 1/2 cup of granulated sugar until it begins to turn an amber color. Add 2 tablespoons of butter and whisk until melted. Slowly pour in the root beer, whisking constantly as it will bubble vigorously. Stir in the heavy cream and a pinch of salt. Simmer for 2-3 minutes, whisking continuously, until the sauce is smooth and slightly thickened.
06 - Remove the bread pudding from the oven and let it rest for 10 minutes. Drizzle generously with the warm root beer caramel sauce just before serving.