Pre-heat the oven to 410°F.
After removing the pits from the cherries, divide them into thirds. Halve one-third of them, quarter another third, and leave the remaining third whole.
In a large bowl, gently mix together the cherries, lemon juice, sugar, corn starch, and both extracts. Set aside.
Roll the first disc of dough out on a floured surface to a 13-inch diameter. Gently roll the dough over your rolling pin and then unroll over the pie dish. The dough should overhang the sides of the dish.
Transfer the cherry mixture into the pie dish with the dough.
Roll the 2nd dough disc out on a floured surface to a 12 to 13-inch diameter. Use a pizza cutter or sharp knife to cut the dough into 10 to 11 1-inch strips.
Place 5 to 6 strips over the top of the pie, spacing them slightly. Fold every other strip all the back to the edge of the dish. Place one of the remaining strips perpendicular to the strips on the pie, right next to where the folded strips are folded back. Return the folded strips back to their normal position. Fold back the other strips and continue until the pie is covered with the dough in a lattice formation. (See photo and/or video for reference).
Use a small sharp knife or kitchen shears to cut away the excess dough hanging over the edges. Be sure to leave enough dough that you can use to crimp the edges (see video for reference).
Make an egg wash by mixing together the beaten egg with the water or milk in a small bowl. Brush the egg wash all over the top layer of dough. Sprinkle the turbinado sugar over the top of the pie.
Bake for 20 minutes, rotating the pie once. Lower the oven temperature to 350°F and bake for another 35 to 40 minutes, or until the crust is golden on top. Cover the edges of the pie with foil or a pie crust shield if getting to brown.
Remove from oven and allow to cool before serving, usually 3 to 4 hours, or overnight.