Step 1: Prepare the Pie Crust: In a large bowl, whisk together 2 ½ cups flour and 1 teaspoon salt. Cut in the cold, cubed butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 2: Add ice water, 1 tablespoon at a time, mixing lightly with a fork until the dough just comes together. Be careful not to overmix. Divide the dough into two equal discs, wrap tightly in plastic, and refrigerate for at least 30 minutes.
Step 3: Prepare the Apple Filling: In a large bowl, combine the sliced apples, granulated sugar, brown sugar, 2 tablespoons flour, cinnamon, nutmeg, cloves, and lemon juice. Toss gently to coat all the apple slices evenly.
Step 4: Preheat Oven: Preheat your oven to 425°F (220°C).
Step 5: Roll Out Bottom Crust: On a lightly floured surface, roll out one disc of dough into a 12-inch circle. Carefully transfer it to a 9-inch pie plate. Trim the edges, leaving about a ½-inch overhang.
Step 6: Add Filling: Pour the apple filling into the pie crust, mounding it slightly in the center. Dot the top of the filling with the small pieces of butter.
Step 7: Roll Out Top Crust: Roll out the second disc of dough into an 11-inch circle. Place it over the apple filling. Trim the edges to match the bottom crust, then crimp the edges together to seal the pie. Cut several slits in the top crust to allow steam to escape.
Step 8: Egg Wash and Bake: Brush the top crust with the beaten egg and sprinkle with coarse sugar. Place the pie on a baking sheet (to catch any drips). Bake for 15 minutes at 425°F (220°C).
Step 9: Reduce Heat and Continue Baking: Reduce the oven temperature to 375°F (190°C) and continue baking for another 45-55 minutes, or until the crust is golden brown and the filling is bubbling. If the edges brown too quickly, cover them with aluminum foil.
Step 10: Cool: Remove the pie from the oven and let it cool on a wire rack for at least 2-3 hours before slicing and serving. This allows the filling to set properly.