In a large bowl, whisk together 1 1/2 cups all-purpose flour, 1/2 teaspoon salt, and 1/2 tablespoon granulated sugar. Add 9 tablespoons cold cubed salted butter and cut it into the flour with a pastry cutter or your fingertips until the mixture resembles coarse meal with pea-sized butter pieces. Drizzle in 4 to 6 tablespoons ice cold water, 1 tablespoon at a time, tossing with a fork until the dough just comes together. Form into a disk, wrap tightly in plastic wrap, and refrigerate for at least 2 hours or up to overnight.

In a medium heavy-bottomed saucepan over medium heat, melt 1 cup granulated sugar, stirring occasionally, until it turns a deep amber color (about 6 to 8 minutes). Off the heat, whisk in 6 tablespoons cubed salted butter until melted (the mixture will bubble). Slowly pour in 1/2 cup plus 1 tablespoon heavy cream while whisking constantly until smooth. Stir in a pinch of flaky sea salt. Let cool slightly; you will use 1/3 cup in the filling and reserve the rest for serving.

On a lightly floured surface, roll the chilled dough into a 12-inch circle about 1/8-inch thick. Gently transfer to a 9-inch pie pan, pressing into the bottom and up the sides. Trim and crimp the edges as desired. Place the crust in the freezer for 15 to 30 minutes while you preheat the oven to 400°F and prepare the filling.

In a large bowl, toss together 8 cups peeled and sliced apples (about 1/4-inch thick) with 1 tablespoon lemon juice, 1/3 cup granulated sugar, 1/3 cup packed light brown sugar, 2 teaspoons cinnamon, 1 teaspoon nutmeg, 1/4 teaspoon cloves, 1/4 teaspoon salt, 1/8 teaspoon allspice, and 1/3 cup all-purpose flour. Let sit for 10 minutes, then stir in 1/3 cup of the reserved caramel sauce.

In a medium bowl, whisk together 1 cup all-purpose flour, 1 cup old-fashioned rolled oats, 2/3 cup packed light brown sugar, 1/4 teaspoon cinnamon, and a pinch of salt. Add 1/2 cup cold cubed salted butter and cut it into the dry ingredients with a pastry cutter until the mixture is crumbly and all the butter pieces are coated.

Beat the egg in a small bowl. Remove the pie crust from the freezer and brush the edges with the beaten egg, then sprinkle with coarse sugar. Spoon the apple filling into the crust, leaving behind any excess liquid. Sprinkle the crumb topping evenly over the apples. Place a baking sheet on the lower rack of the oven to catch any drips.

Bake at 400°F for 45 to 55 minutes. Check at 25 to 30 minutes; if the crust or topping is browning too quickly, cover loosely with foil or a pie shield. The pie is done when the filling is bubbly, the topping and crust are golden brown, and a fork inserted through the topping meets tender apples.

Let the pie cool at room temperature for 2 to 3 hours before slicing. Serve with extra caramel sauce. Store leftovers covered in the refrigerator for up to 5 days.

Make ahead: The crust can be made 3 days ahead (refrigerated) or frozen for 2 months. The baked pie can be made 1 to 2 days ahead; cool completely, cover with foil, and refrigerate. Rewarm at 350°F, loosely covered, for 12 to 20 minutes before serving.