In a mixing bowl, combine 2 ½ cups flour, ½ teaspoon salt, and 1 tablespoon sugar. Add the chilled, cubed butter and use a pastry cutter or your fingers to blend until the mixture resembles coarse crumbs.

Gradually add ice water, one tablespoon at a time, mixing gently until the dough just comes together. Divide the dough in half, shape into disks, wrap in plastic wrap, and refrigerate for at least 1 hour.

Preheat the oven to 425°F (220°C).

In another bowl, mix the sliced peaches, ¾ cup granulated sugar, ¼ cup brown sugar, cornstarch, cinnamon, nutmeg, lemon juice, and vanilla extract. Stir until well combined and let sit.

Roll out one disk of dough on a floured surface to fit your 9-inch pie dish. Carefully place the crust in the pie dish, trimming the edges as necessary.

Pour the peach filling into the prepared crust, spreading it evenly.

Roll out the second disk of dough and place it over the peach filling. You can cut slits in the top for a lattice effect or simply judge for an entire cover. Trim and crimp the edges to seal.

In a small bowl, whisk together the egg and milk for the egg wash. Brush the top crust with the egg wash to give it a golden finish.

Bake in the preheated oven for 45 minutes, or until the crust is golden brown and the filling is bubbly. If the edges brown too quickly, cover them with foil.

Remove from the oven and let the pie cool for at least 2 hours before slicing. This allows the filling to set.