In a large bowl, combine the peaches and sugars. Cover and let sit at room temperature for 1 hour.

Preheat the oven to 400°F. Place a foil-lined rimmed baking sheet on the lower third of the oven to preheat.

On a floured surface, roll out 1 piece of dough into a 12-inch round, starting at the center and working your way out. (Sprinkle flour over the top of the dough if it's a bit too moist.) Carefully lift the dough and place in a 9-inch pie pan (not deep dish); gently press against the sides of the pan. Trim excess overhanging dough at the edge of the pie plate. Cover and refrigerate until ready to use.

On a floured surface, roll out the other piece of dough into a 12-inch round. Cut into sixteen (¾-inch-wide) strips. Place the dough on a baking sheet, cover and refrigerate until ready to use.

Drain the peaches, reserving ½ cup of the liquid. Discard the remaining liquid. Remove 1 cup of the peaches to a medium bowl and mash with a fork.

In a medium saucepan, combine the cornstarch, cardamom, salt, and ginger. Gradually whisk in the peach juice. Bring to a boil over medium heat, stirring constantly to prevent lumps. Use a rubber spatula to scrape the bottoms and sides of the saucepan. Reduce the heat to medium low and cook until thickened, 2 to 3 minutes. Whisk in the butter, lemon zest, and lemon juice. Fold in the peaches and mashed peaches. Pour the mixture into the chilled crust.

Arrange 8 strips of dough in parallel rows on top of the filling. Working with one strip at a time, arrange the remaining 8 strips perpendicular to the first ones, weaving the strips over and under to form a lattice. Trim the ends of the strips and press them against the bottom crust. Crimp as desired.

In a small bowl, whisk the egg and 1 tablespoon of water with a fork. Brush the dough lightly with the egg wash and sprinkle with the turbinado sugar.

Bake the pie until the crust is set and beginning to brown, 20 to 25 minutes. Rotate the pan and reduce the oven temperature to 375°F. Continue to bake until the filling is bubbly and thickened in the middle of the pie, 45 to 55 minutes more. (Cover the pie with foil if the crust is getting too dark after 55 minutes.)

Allow the pie to cool to room temperature before slicing, about 4 hours.