Mix diced apple, cold water, and lemon juice together in a small bowl; soak for 10 minutes. Drain.
Stir cream cheese, brown sugar, cinnamon, nutmeg, and allspice together in a bowl until smooth. Fold the diced apple into the cream cheese mixture.
Spoon the cream cheese mixture onto the center of a sheet of plastic wrap. Use the plastic to shape the cheese mixture into a log, wrap the plastic around the log, and refrigerate until set, about 2 hours.
Cook and stir white sugar in a 3-quart saucepan over medium-high heat until melted, smooth, and taking on an amber color, about 10 minutes; add butter and stir quickly with a whisk until the butter is melted, 1 to 2 minutes. Immediately remove saucepan from heat. Slowly stream heavy cream into the sugar mixture while continually stirring with the whisk to incorporate fully; add sea salt and stir. Transfer the caramel sauce to a glass bowl and set aside to cool.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Roll out pie pastry on a lightly floured surface to about 1/4-inch thick. Cut dough into rectangles about 2 inches long. Arrange dough strips onto the prepared baking sheet. Brush the tops of the dough strips with beaten egg. Sprinkle cinnamon-sugar over the strips.
Bake dough strips in preheated oven until golden brown, 10 to 15 minutes. Cool completely.
Spread pecans into a shallow dish. Unwrap cheese log and roll in the pecans to coat. Place coated log onto a plate and drizzle caramel sauce evenly over the log. Serve with the pie crust chips.