Preheat oven to 375°F.

Prepare filling: Grate apple on the large side of a box grater. Squeeze out excess juices from grated apples (don’t dry them out completely, but they shouldn’t be dripping). Dice other half of apple into very small (about 1/8") chunks.

Melt butter in a saucepan over low heat. Add grated and diced apples and stir to coat. Add flour, cinnamon, and salt and cook over medium heat, about 5 minutes. Add caramel sauce and cook for another 5 minutes, until thick and syrupy. Let cool before using.

On a well-floured surface, roll out one sheet of puff pastry into a large 10”-x-14” rectangle. Using a pizza cutter or very sharp knife, cut rectangle in half lengthwise, creating two 5”-x-14” rectangles. Cut each into four equal rectangles.

Place one heaping tablespoon caramel apple mixture in the middle of four rectangles of puff pastry. Brush the other four rectangles of puff pastry with beaten egg using a pastry brush and lay egg-washed side down on top of filling. Use a fork to seal edges and make a few holes with tines of the fork on top. Repeat with other sheet of puff pastry.

Bake strudels on a parchment-lined baking sheet for 30-25 minutes or until light golden brown. Let cool 15 minutes before serving. Best served warm. Reheat for a few minutes in a toaster oven if desired.

For icing: In a mixing bowl, beat together cream cheese, powdered sugar, and vanilla extract until smooth. Transfer to a plastic bag or piping bag and drizzle on toaster strudels just before serving.