Step 1: In a large bowl, whisk together 2½ cups flour and 1 teaspoon salt. Add the cold, cubed butter and cut it into the flour using a pastry blender, two knives, or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 2: Gradually add ½ cup ice water, one tablespoon at a time, mixing gently until the dough just comes together. Be careful not to overmix. If the dough is too dry, add a little more water, half a tablespoon at a time. Divide the dough in half, flatten each into a disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes (or up to 2 days).
Step 3: While the dough chills, prepare the apple filling. In a large bowl, combine the sliced apples, granulated sugar, brown sugar, 1 tablespoon flour (or cornstarch), cinnamon, nutmeg, cloves (if using), and lemon juice. Toss gently to coat the apples evenly. Set aside.
Step 4: Preheat your oven to 425°F (220°C). On a lightly floured surface, roll out one disc of chilled dough into a 12-inch circle. Carefully transfer the dough to a 9-inch pie plate. Trim the edges, leaving about a ½-inch overhang.
Step 5: Pour the apple filling into the pie crust, mounding it slightly in the center. Dot the top of the apples with the 2 tablespoons of small butter pieces.
Step 6: Roll out the second disc of dough. Place it over the apple filling. You can cut slits in the top for steam or create a lattice design. Fold the overhanging bottom crust edge over the top crust edge, then crimp the edges together using your fingers or a fork to seal.
Step 7: In a small bowl, whisk together the egg yolk and 1 tablespoon water for the egg wash. Brush the top crust evenly with the egg wash, then sprinkle with coarse sugar.
Step 8: Place the pie on a baking sheet to catch any drips. Bake for 15 minutes at 425°F (220°C). Reduce the oven temperature to 375°F (190°C) and continue baking for another 45-60 minutes, or until the crust is deeply golden brown and the filling is bubbling thickly. If the crust edges brown too quickly, loosely cover them with aluminum foil.
Step 9: Remove the pie from the oven and let it cool completely on a wire rack for at least 3-4 hours before slicing and serving. This allows the filling to set properly.