Spray 9" x 13" baking dish with nonstick cooking oil. Cube stale bread and add it to the dish.
Combine condensed milk, coconut milk, eggs, vanilla extract, brown sugar, cinnamon, and sea salt. Whisk until consistent and pour over cubed bread, then set aside.
Peel, core, and chop apples into cubes, then add to a pot along with butter, brown sugar, and pumpkin pie spice. Mix and cook over low-medium heat until butter melts and apples are evenly coated, then add water and boil for 5 minutes or until apples are slightly softened.
Strain apple mixture, then distribute over the top of the bread pudding. Place the baking dish, uncovered, on the middle rack in a preheated oven and bake for 45 minutes to an hour, or until set.
Add butter and brown sugar to a saucepan, stirring over low heat until butter melts and mixture is consistent (about 2 minutes). Add heavy cream, salt, and vanilla, then mix another 5 minutes over medium heat until sauce is creamy and pourable.
Drizzle caramel sauce over the bread pudding, then serve.