Step 1: For the crust: In a large bowl, whisk together the all-purpose flour, 1 teaspoon granulated sugar, and 1 teaspoon salt.
Step 2: Add the very cold, cubed butter to the flour mixture. Using a pastry blender or your fingertips, cut in the butter until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 3: Gradually add ice water, one tablespoon at a time, mixing lightly with a fork until the dough just comes together. Do not overmix. You might not need all the water.
Step 4: Divide the dough in half, flatten each half into a disk, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes.
Step 5: For the filling: In a large bowl, gently combine the sliced peaches with 3/4 cup granulated sugar, cornstarch (or flour), ground cinnamon, ground nutmeg, lemon juice, and a pinch of salt. Toss everything together until the peaches are evenly coated. Let the mixture sit for about 10-15 minutes.
Step 6: Preheat your oven to 400°F (200°C).
Step 7: On a lightly floured surface, roll out one disk of chilled pie dough into a 12-inch circle. Carefully transfer the dough to a 9-inch pie plate. Trim the edges, leaving about a 1/2-inch overhang.
Step 8: Pour the peach filling into the pie crust, mounding it slightly in the center. Dot the top of the filling with the small pieces of butter.
Step 9: Roll out the second disk of pie dough. Place it over the filling, or cut it into strips for a lattice top. Trim excess dough and crimp the edges of both crusts together to seal.
Step 10: Brush the top crust with the egg wash and sprinkle with the remaining 1 tablespoon of granulated sugar.
Step 11: Place the pie on a baking sheet and bake for 20 minutes at 400°F (200°C).
Step 12: Reduce the oven temperature to 375°F (190°C) and continue baking for another 40-50 minutes, or until the crust is deeply golden brown and the filling is bubbling thickly. If the crust starts browning too quickly, loosely cover the edges with aluminum foil.
Step 13: Remove from the oven and let cool completely on a wire rack for at least 3-4 hours before slicing and serving. This allows the filling to set properly.