Score an X on the bottom of each peach, blanch in boiling water for 30-45 seconds, then transfer to ice water. Peel skins off, then slice into 1/4 to 1/2 inch thick pieces to get 6 cups.
Toss the sliced peaches with granulated sugar, brown sugar, and lemon juice. Let sit at room temperature for 20-30 minutes to release juices, then drain, reserving both the peaches and the liquid separately.
Whisk cornstarch into the reserved peach liquid. Cook over medium heat, stirring constantly, until thickened into a glossy syrup (3-4 minutes). Remove from heat, stir in cinnamon, nutmeg, salt, and vanilla, then fold back into the drained peaches.
On a floured surface, roll one crust into a 1-2 inch wider circle. Press into the pie dish without stretching, trim overhang to 1 inch, and refrigerate while preparing the rest.
Pour the peach filling into the chilled crust, mounding it slightly higher in the center. Scatter the butter pieces evenly over the top.
Roll out the second crust and place it over the filling. Trim and crimp the edges together. Cut 4-5 slits in the top crust for steam to escape.
Brush the top crust with beaten egg and sprinkle with turbinado sugar if using. Refrigerate or freeze the assembled pie for 15 minutes to firm up the crust.
Preheat oven to 425°F (220°C). Place pie on a baking sheet and bake for 20 minutes. Reduce heat to 375°F (190°C) and bake for 35-40 minutes more, using a crust shield if edges brown too quickly, until crust is golden and filling is bubbling.
Transfer the pie to a wire rack and let cool at room temperature for at least 3 hours before slicing to allow the filling to set properly.