In a large bowl, combine the flour, salt, and sugar.
Add the cold cubed butter and work it into the flour until the mixture is crumbly with small pieces of butter still visible.
Add the ice water gradually, 1 tablespoon at a time, until the dough just comes together. Divide into 2 discs and chill for 30 minutes.
In another bowl, combine the sliced peaches with the sugar, cornstarch, cinnamon, optional nutmeg, vanilla, and lemon juice. Toss until evenly coated.
Preheat the oven to 425°F (220°C).
Roll out one disc of dough and fit it into a 9-inch pie pan without stretching the pastry.
Spoon the peach filling into the crust and spread it evenly.
Dot the filling with 1 tablespoon butter.
Roll out the remaining dough and cut it into 3/4- to 1-inch-wide strips.
Weave the strips over the peaches to create a lattice, then trim and seal the edges.
Beat the egg with 1 tablespoon water.
Brush the lattice lightly with egg wash and sprinkle with coarse sugar, if using.
Bake for 15 minutes at 425°F (220°C), then reduce the oven to 375°F (190°C) and bake for another 40–45 minutes, until the crust is golden and the peach filling is bubbling.
Transfer the pie to a wire rack and cool for at least 3 hours before slicing so the filling can set.