1. Drizzle the lemon juice over the apples and toss to coat. Set the apples aside.
2. Stir together the sugars, cornstarch, cinnamon, nutmeg and salt in a large saucepan. Whisk in the water until everything has dissolved.
3. Bring the mixture to a boil over medium-high heat. Reduce the heat slightly and simmer for 2 minutes.
4. Add the apples and increase the heat to medium-high heat to bring the mixture to a full boil again. Reduce the heat to low heat and simmer, stirring occasionally, for 10-12 minutes. Remove the pan from the heat and let the mixture cool.
5. Refrigerate the filling in an airtight container for 4-6 hours to thicken. Store in the refrigerate for up to 2 weeks.

TO MAKE THE PIE:

6. Preheat the oven to 375°F. Spray a 9-inch deep dish pie plate with non-stick baking spray.
7. Place the pie crust in the pie plate. Fold the edges of the crust under and mold them to your preference. Refrigerate until ready to use.
8. To make the cheesecake layer, beat the cream cheese until creamy.
9. Add the sugar and beat the mixture until smooth.
10. Add the egg and beat until full incorporated. Do not overbeat the batter.
11. Spread the cheesecake batter in the prepared pie crust.
12. Gently spoon the apple pie filling on top of the cheesecake batter.
13. For the crumb topping, mix together the crumb topping ingredients. Sprinkle the mixture evenly over the apple pie filling.
14. Bake the cheesecake pie for 40 minutes.
15. Remove the pie plate from the oven and place it on a wire rack. Let the cheesecake pie cool for 1 hour, then refrigerate for 2-3 hours, until completely chilled.
16. Serve with vanilla ice cream and caramel topping, if desired.