When ready to make a pie, remove 1 disk of dough from the refrigerator. Allow it to sit for up to 10 minutes, if necessary, to become pliable enough to roll.
Cover the work surface with a large pastry cloth and lightly dust with flour. Use a rolling pin to roll the dough to a circle about 1/8 inch thick by 12 inches in diameter. The dough should be smooth and cool. Fold it carefully in half, then again into quarters and transfer to a 9-inch pie plate. Unfold the dough and trim the border just short of the edge of the plate. Cover with plastic wrap and refrigerate for at least 30 minutes and up to 3 hours.
Next, make the lattice top. Roll out the other disk into a 12-inch circle about 1/8 to 1/4-inch thick. Cut into strips. Transfer the strips to a baking sheet and refrigerate while you prepare the filling.
Peel and core 2 1/2 pounds of pears and slice them into 1/4-inch slices. Transfer the slices to a large bowl along with 1/4 cup dark brown sugar, 1/4 cup cornstarch, and 1 teaspoon King Arthur Flour chai spice mix. Stir gently to combine.
Add half the pears to the pie shell, and drizzle 1/4 cup of the maple syrup over the top, then add the remaining pears. Drizzle the remaining caramel on top. Lay out the prepared pie dough strips for the lattice and carefully weave a lattice. Trim off any excess lattice too long for the pie plate and fold the edges of the bottom crust up and over the lattice strips. Crimp the edges together. Slide the whole pie into the freezer until the crust is very firm, about 15 minutes, before baking.
In the meantime, center a rack in the oven and preheat to 350 (F). Prepare a rimmed baking sheet by lining with parchment paper.
When the pie dough is firm, it's ready to be baked. Beat 1 large egg and 1 teaspoon in a small bowl, then use a pastry brush to brush the top of the pie with egg wash. Sprinkle the egg wash with demarara sugar. Bake the pie until it is deep golden brown and the juices are bubbling, about 1 hour. If the crust begins to burn before the filling bubbles, tent it with aluminum foil. Cool on a wire rack slightly before serving. The pie is best the day it's made.