Prepare the Dough: In a large bowl, whisk together the flour, 1 tablespoon sugar, and salt. Cut in the frozen butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining. Gradually drizzle in ice water, a tablespoon at a time, mixing until the dough just comes together. Divide the dough in half, shape each into a disk, wrap in plastic, and chill for at least 1 hour.

Prepare the Filling: In a separate large bowl, gently toss the sliced peaches with the granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, and lemon juice. Set aside while you roll out the bottom crust.

Assemble the Pie: Preheat oven to 425°F (220°C). On a lightly floured surface, roll out one disk of dough into a 12-inch circle. Carefully transfer the dough to a 9-inch pie plate. Trim the edge, leaving a 1-inch overhang.

Fill and Cover: Pour the peach filling into the bottom crust. Dot the top of the peaches evenly with the 2 tablespoons of butter pieces. Roll out the second disk of dough and place it over the filling. Trim the top crust overhang to 1 inch. Crimp the edges of the top and bottom crusts together to seal. Cut several slits in the top crust to allow steam to escape.

Bake the Pie: Whisk together the egg and milk/cream for the egg wash and brush lightly over the top crust. Sprinkle lightly with extra sugar if desired. Place the pie on a baking sheet (to catch drips) and bake at 425°F (220°C) for 15 minutes. Then, reduce the oven temperature to 375°F (190°C) and continue baking for another 45-50 minutes, or until the crust is golden brown and the filling is visibly bubbly.

Cool Completely: Let the pie cool on a wire rack for at least 3-4 hours before slicing. This allows the filling to set properly.