In a large bowl, combine the apples with 1 cup of sugar. Stir well and set aside to allow the apples to release some of their natural juices.
In another bowl, whisk together the flour, salt, 1 tablespoon of sugar, and melted butter. Mix until crumbly, ensuring everything is evenly distributed.
Separate the egg, placing the yolk in a 1 cup measuring cup. Add milk until it reaches the 2/3 line and whisk until well combined.
Pour the egg and milk mixture into the flour mixture, stirring until a dough forms. It should be soft and slightly sticky, but manageable.
Take half of the dough and roll it out on a floured surface to fit a 10x15 inch jelly roll pan. This will be the bottom layer of your bars.
Place the rolled-out dough into the bottom of the jelly roll pan, pressing gently to fit the corners.
Sprinkle the crushed Rice Krispies evenly over the dough layer to provide a delightful crunch.
Top the cereal layer with the apple mixture, spreading it out evenly to ensure every bite is filled with fruit.
Roll out the remaining dough and place it over the apple filling, sealing the edges gently.
Beat the reserved egg white until foamy and brush it over the top of the bars. Sprinkle a little sugar for added sweetness and texture.
Bake in a preheated oven at 350 degrees Fahrenheit for 45 minutes. The top should be golden brown and the apples bubbling beneath.
While the bars are cooling, prepare the frosting. In a microwave-safe bowl, combine the cream cheese and butter, microwaving in 15 to 20 second intervals until soft enough to whisk.
Add the caramel ice cream topping, milk, and vanilla extract, whisking until smooth and creamy.
Gradually incorporate the powdered sugar into the mixture until it reaches a smooth consistency. Drizzle this luscious frosting over the cooled bars for a sweet finish.
