Prepare the Crust: In a large bowl, whisk together flour and salt. Cut in the cold butter using a pastry blender or your fingers until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.

Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together. Divide the dough in half, flatten into discs, wrap in plastic, and refrigerate for at least 30 minutes.

Prepare the Filling: Whisk together the sugar, cornstarch, and nutmeg in a large bowl. Add the drained canned peaches, reserved syrup (if using 1/4 cup of it), lemon juice, and vanilla extract. Gently toss until the peaches are evenly coated.

Assemble the Pie: Preheat your oven to 400°F (200°C). On a lightly floured surface, roll out one disc of dough and carefully place it in a 9-inch pie dish. Pour the peach filling into the bottom crust. Dot the top of the filling with the 1 tablespoon of small butter pieces.

Cover the pie with the second crust, trim the edges, crimp them together, and cut several slits in the top crust to allow steam to escape. Brush the top crust lightly with the beaten egg wash.

Bake for 15 minutes at 400°F (200°C). Then, reduce the oven temperature to 375°F (190°C) and continue baking for an additional 40-45 minutes, or until the crust is golden brown and the filling is bubbling thickly. If the edges brown too quickly, cover them with foil.

Cool the pie completely on a wire rack for at least 3-4 hours before slicing and serving to allow the filling to set properly.