Step 1: Prepare the Pie Crust. In a large bowl, whisk together 2½ 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 butter pieces. Gradually add ice water, 1 tablespoon at a time, mixing until the dough just comes together. Do not overmix. Divide dough in half, flatten into discs, wrap tightly in plastic, and chill for at least 30 minutes in the refrigerator.
Step 2: Brown the Butter. In a small, light-colored saucepan, melt ½ cup unsalted butter over medium heat. Continue cooking, stirring occasionally, until the butter turns a rich golden-brown and emits a nutty aroma. Be careful not to burn it. Remove from heat and set aside to cool slightly.
Step 3: Prepare the Peach Filling. In a large bowl, combine the sliced peaches, granulated sugar, brown sugar, maple syrup, ¼ cup flour, cinnamon, nutmeg, and a pinch of salt. Gently stir until the peaches are well coated. Pour in the slightly cooled brown butter and mix gently to combine everything evenly.
Step 4: Assemble the Pie. On a lightly floured surface, roll out one disc of chilled pie dough into a 12-inch circle. Carefully transfer the dough to a 9-inch pie plate. Trim the edges, leaving a ½-inch overhang. Pour the prepared peach filling evenly into the crust.
Step 5: Create the Top Crust. Roll out the second disc of dough. You can either place it as a solid top crust (cutting several slits in the center for steam to escape) or cut it into strips to create a lattice top. Place the top crust over the filling. Trim any excess dough and crimp the edges with the bottom crust to seal.
Step 6: Chill and Preheat. Place the assembled pie in the refrigerator for 15 minutes. Meanwhile, preheat your oven to 400°F (200°C).
Step 7: Egg Wash and Bake. In a small bowl, whisk together the beaten egg and milk for the egg wash. Brush the top crust of the chilled pie with the egg wash and generously sprinkle with coarse sugar. Place the pie on a baking sheet to catch any potential drips.
Step 8: Bake the Pie. Bake for 15 minutes at 400°F (200°C), then reduce the oven temperature to 375°F (190°C) and continue baking for another 35-45 minutes, or until the crust is deeply golden brown and the filling is bubbling thickly. If the crust edges start to brown too quickly, cover them loosely with aluminum foil.
Step 9: Cool and Serve. Transfer the baked pie to a wire rack and let it cool for at least 2-3 hours before slicing and serving. This crucial step allows the filling to set properly. Serve warm, perhaps with a scoop of vanilla ice cream.