Combine the flour, sugar, and salt in a large bowl. Add the cold butter pieces and use a pastry cutter or your fingertips to work the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces remaining. Gradually add ice water, one tablespoon at a time, mixing gently until the dough begins to come together.
Divide the dough into two equal portions and shape each into a disk. Wrap each disk in plastic wrap and refrigerate for at least 30 minutes or up to 2 days.
In a large bowl, gently toss the sliced peaches with lemon juice. In a separate bowl, mix together the sugars, cornstarch, cinnamon, nutmeg, and salt. Add this dry mixture to the peaches and toss until evenly coated.
Preheat your oven to 425°F (220°C). Roll out one disk of dough on a lightly floured surface into a 12-inch circle. Transfer to a 9-inch pie dish, letting the excess hang over the edges. Pour the peach filling into the crust and dot with butter pieces.
Roll out the second disk of dough into another 12-inch circle. You can either cover the pie completely or create a lattice top for a more decorative presentation. If covering completely, cut several slits for steam to escape. Trim and crimp the edges to seal.
Brush the top crust with the egg wash and sprinkle with coarse sugar. Place the pie on a baking sheet (to catch any drips) and bake at 425°F for 20 minutes. Reduce the temperature to 375°F (190°C) and continue baking for 30-35 minutes until the crust is golden and the filling is bubbling.
Allow the pie to cool on a wire rack for at least 2 hours before slicing. This cooling period is crucial as it allows the filling to set properly.