Pulse the flour, sugar, and salt in a food processor until combined.
Add the shortening and process until the mixture has the texture of coarse sand; about ten one-second pulses.
Scatter the butter pieces over the flour mixture and process until the mixture is pale yellow and resembles coarse crumbs, with butter a bit larger than small peas; about ten one-second pulses.
Blend in three tablespoons ice water until dough just comes together.
(Add up to one tablespoon more ice water if dough does not come together.)
Turn the dough out onto a sheet of plastic wrap and flatten it into a 4-inch disk.
Wrap the dough and refrigerate it for 30 minutes (or up to 2 days) before rolling. (If the dough is refrigerated longer than one hour, let it sit out at room temperature until it's malleable.)
For the filling:
Adjust the oven rack to the upper-middle position (between 7 and 9 inches from heating element) and heat the oven to 500°F.
Whisk the cider, syrup, lemon juice, cornstarch, and cinnamon (if using) together in a medium bowl until smooth.
Heat the butter in a 12-inch ovenproof pan over medium-high heat. When the foaming subsides, add the apples and cook, stirring two or three times until the apples begin to caramelize, about five minutes. (Do not fully cook the apples.)
Remove the apples from the heat, add the cider mixture, and gently stir until the apples are well coated.
Set aside in the pan to cool slightly (the pie will bake in this pan).
Assemble and bake:
Roll out the dough between layers of waxed paper to an 11-inch circle. Place the dough on the pan over the apple filling.
Brush the dough with egg white and sprinkle with sugar.
With a sharp knife, gently cut the dough into six pieces by making one vertical cut followed by two evenly spaced horizontal cuts (perpendicular to the first cut).
Bake until the apples are tender and the crust is a deep golden brown, about 20 minutes.
Let the pie cool 15 minutes; serve.
August 6, 2008 Awesome! Left out the cinnamon and cooked the apples for a nice long time beforehand. Recipe from Cook’s Illustrated.