Using a pastry brush, spread 2 tablespoons of unsalted butter onto the sides and bottom of a 9x13 baking dish. Set aside.
In a large mixing bowl, whisk together the eggs, half & half, light brown sugar, vanilla extract, apple pie spice blend, and salt. Be sure that all the eggs are well beaten, and all the ingredients are well combined.
Into the bottom of the prepared 9x13 baking dish, place an even layer of half of the cubed challah bread.
Sprinkle half of the diced spiced apples evenly over the top of the layer of cubed challah bread.
Repeat with a second layer of cubed challah bread and diced spiced apples.
Slowly pour the egg mixture over the bread and apples making sure you cover all the bread with the egg mixture. You can use the back of a large spoon to push any pieces of bread into the egg mixture if they did not get covered when you were pouring it over the bread.
Melt the remaining 4 tablespoons of unsalted butter and drizzle it over the entire top of the apple bread pudding.
Cover the baking dish with aluminum foil and refrigerate for a minimum of 4 hours or up to overnight.
Preheat the oven to 350°F. While your oven is preheating, you will want to allow your baking dish to sit at room temperature to warm up before placing it into a hot oven. This is especially important if you are using a glass baking dish.
Bake the apple bread pudding for 45 minutes, covered with aluminum foil. Remove the foil and bake for an additional 30 minutes or until the top is lightly golden and a knife inserted into the center comes out clean.
While your apple bread pudding finishes baking, you can make your caramel sauce by adding the unsalted butter, granulated sugar, and light brown sugar to a saucepan on medium heat. You will melt your butter and cook your sugars for 5 to 6 minutes, whisking constantly or until the mixture reaches 220°F on a candy thermometer. Be very careful not to let your sugars burn.
Remove the saucepan from the heat and whisk in the half & half, vanilla extract, and salt until fully incorporated. Be very careful working with hot sugars like this caramel sauce. It can cause severe burns if not handled properly.
When your apple bread pudding comes out of the oven, you will slowly pour the caramel sauce over the top of the warm bread pudding. Be sure to reserve about ½ cup of the caramel sauce to serve over individual servings of the apple bread pudding. The caramel sauce will make approximately 1½ cups of sauce.