Make the crust: In a large bowl, whisk together flour and salt. Add the cold butter cubes and cut them into the flour with a pastry cutter or fingertips until most pieces are pea-sized with some flat, floury flakes.
Sprinkle in 1/3 cup ice water and toss with a fork. Add more water a tablespoon at a time just until the dough holds together when pressed. Lightly smear portions of the dough forward on the counter once or twice (fraisage) to create thin buttery layers. Divide into two discs, wrap, and chill 45–60 minutes.
While the dough chills, prepare the peaches: In a large bowl, combine sliced peaches with brown sugar, granulated sugar, lemon juice, almond extract, cinnamon, and a pinch of salt. Toss well and let stand 20 minutes to draw out the juices.
Roll the first dough disc on a lightly floured surface to a 12-inch round. Fit it into a 9-inch pie plate, letting about 1 inch overhang. Refrigerate the lined pan for 15 minutes to keep the bottom crust firm.
Set a rack in the lower third of the oven and preheat to 400°F (204°C). Place a foil-lined baking sheet on the rack to catch drips during baking.
Drain the macerated peaches through a sieve set over a small saucepan. Bring the collected juices to a simmer over medium heat and cook 3–5 minutes until syrupy and reduced by about half. Remove from heat and cool 1–2 minutes. Toss the peaches with the flour, then fold in the warm syrup until evenly coated.
Spoon the filling into the chilled pie shell and dot the top with the diced butter.
Roll the second dough disc to an 11-inch round. Lay it over the filling, trim excess, tuck the edges under, and crimp. Cut 4–5 vents in the top crust. Brush with egg wash and sprinkle with turbinado sugar. Freeze the assembled pie for 10 minutes to firm the pastry.
Bake on the preheated sheet for 25 minutes at 400°F, then reduce the oven temperature to 365°F (185°C) and continue baking 35–45 minutes, until the crust is deeply golden and the juices bubble thickly through the center vents. Shield the edges with foil if they brown too quickly.
Cool the pie on a wire rack for at least 3 hours before slicing so the juices set. Serve slightly warm or at room temperature.