In a large bowl, combine the pitted and halved cherries, sugar, cornstarch, lemon juice, vanilla extract, salt, and almond extract (if using). Stir until the cornstarch is fully incorporated and no white streaks remain. Let the mixture sit at room temperature while you prepare the crust; the cherries will begin to release their juices.

In a food processor fitted with the steel blade, pulse together the flour, sugar, and salt. Add the cold butter cubes and pulse 8 to 10 times, until the largest butter pieces are the size of small peas. (Alternatively, you can make the crust by hand: whisk the dry ingredients in a large bowl, then cut in the butter using a pastry cutter.)

Add the ice-cold water, 1 tablespoon at a time, pulsing after each addition, until the dough just begins to come together. It should look shaggy and slightly crumbly but hold together when pinched.

Turn the dough out onto a lightly floured surface and gather it into a ball. Divide it in half, shape each half into a flat disk, and wrap one disk tightly in plastic wrap. Place the wrapped disk in the refrigerator.

On a lightly floured surface, roll the remaining disk into a 12-inch circle. Carefully transfer it to a 9-inch deep-dish pie plate, gently pressing it into the bottom and up the sides. Trim any excess dough to a 1/2-inch overhang. Place the pie plate in the refrigerator while you roll the top crust.

Remove the chilled bottom crust from the refrigerator. Stir the cherry filling again, then pour it (with all the juices) into the crust. Dot the filling with the small cubes of butter.

Take the remaining dough disk from the refrigerator and roll it out to a 12-inch circle. Place the top crust over the filling. Trim the overhang to 1/2 inch, then fold the edges under and crimp with your fingers or a fork. Cut several steam vents in the top crust.

For a lattice top: Cut the rolled dough into strips about 3/4 to 1 1/4 inches wide. Lay the longest strip across the center of the pie. Place more strips parallel to the first, spaced about 1/2 inch apart. Fold back alternating strips and weave the perpendicular strips over and under to create a lattice. Trim and crimp the edges.

In a small bowl, whisk together the egg and milk. Lightly brush this mixture over the top crust (and lattice strips, if using). Sprinkle with coarse sugar if desired.

Place the pie in the refrigerator for 20 minutes to chill. Meanwhile, preheat your oven to 400°F with a rack positioned in the lower third.

Bake the pie on the lower rack at 400°F for 20 minutes. Then, reduce the oven temperature to 350°F and continue baking for 40 to 50 minutes more, until the filling is bubbling vigorously through the vents. If the crust starts browning too quickly, loosely cover the pie with aluminum foil after about 30 minutes of total baking.

Let the pie cool completely on a wire rack, at least 3 hours, before slicing. This allows the filling to set properly.