Move the oven rack to the middle position and preheat the oven to 425°F.
Place the sliced apples into a large bowl. Add the brown sugar, granulated sugar, flour, cinnamon, nutmeg, salt, vanilla extract, and lemon juice. Toss thoroughly until the apple slices are evenly coated.
6 to 7 Granny Smith and Honeycrisp apples, ½ cup brown sugar, ¼ cup granulated sugar, 2 Tablespoons all-purpose flour, 1 teaspoon ground cinnamon, ¼ teaspoon ground nutmeg, ¼ teaspoon salt, 1 teaspoon pure vanilla extract, 1 Tablespoon fresh lemon juice
Gently fit the bottom crust into a 9-inch pie dish, leaving the excess dough overhanging the rim. Layer half the spiced apples over the bottom crust, then drizzle with ¼ cup salted caramel sauce. Top with the remaining apples and drizzle with the remaining ¼ cup caramel sauce.
½ cup salted caramel sauce
Place the second rolled crust over the apple filling. Trim any excess dough and crimp the edges together to seal the pie. Cut a few small slits in the top crust to allow steam to escape during baking.
Whisk the egg and milk together in a small bowl until smooth. Brush the egg wash evenly over the top crust, then sprinkle the surface with the turbinado sugar.
1 large egg, 1 Tablespoon milk, 1 Tablespoon turbinado sugar