1. Start by making the crust. Mix the flour and salt together in a large bowl.
2. Using a pastry cutter, cut the cold butter into the flour mix until the mixture resembles course meal.
3. Whisk together the egg and water and then add slowly to the flour mixture while stirring with a wooden spoon or rubber spatula.
4. When the dough starts to form, transfer it to a floured work surface and fold the dough until the flour is fully incorporated.
5. Divide the dough in half, flatten each into a 1 inch thick disk, and wrap tightly in plastic wrap.
6. Chill in the refrigerator for at least 2 hours.
7. While the crust is in the refrigerator, make the homemade caramel sauce and prep the apple filling. Find the recipe and instructions for the caramel sauce here.
8. To make the filling, gently toss the thinly sliced apples and lemon juice in a very large bowl.
9. In a small mixing bowl, combine the flour, nutmeg, cinnamon, and salt.
10. Pour over the apples and gently stir to combine. Set aside.
11. Roll out one of the disks of chilled dough on a floured work surface until it is a circle about 12 inches in diameter.
12. Carefully place the dough into a 9x2 inch pie dish.
13. Fill the pie crust with the apples. Drizzle with 1/2 cup of the caramel sauce, reserving the rest for topping.
14. Preheat oven to 375 degrees F.
15. Remove the other disk of pie dough from the refrigerator.
16. Roll the dough out into another 12 inch circle. Using a pizza cutter and a clean ruler, cut 10 strips 1 inch wide.
17. Carefully thread the strips over and under one another, pulling back strips as necessary to weave.
18. Using a sharp knife, trim the extra overhang. Press down the edges of the dough with a fork or your fingers.
19. Brush the lattice crust with a very thin layer of the beaten egg.
20. Sprinkle with a little cinnamon.
21. Place the pie on a large baking sheet and bake for 20 minutes. Cover the edges of the crust with aluminum foil and bake for an additional 40-50 minutes. The pie will be done when the caramel begins to bubble up.
22. Allow the pie to cool for 4 hours before serving.
23. Drizzle each serving with the extra caramel sauce. It is best served on the same day, but it can be covered tightly and stored in the refrigerator for up to 3 days.