In a large mixing bowl, combine flour and salt. Using a pastry blender, cut in the shortening and butter until mixture is uniform and resembles coarse cornmeal.
Sprinkle the cold water over the mixture 1 tablespoon at a time, tossing lightly with a pastry blender. Just only enough water to hold the dough together. The dough must not be sticky—it should come away from the bowl clean.
Divide dough in half, then use your hands to shape the pastry into a ball then flatten it with your palm. Either wrap each half of dough tightly in Seran wrap and place it in the refrigerator overnight at this point, or continue. Between two sheets of lightly floured waxed paper or Seran wrap, use light strokes from the center in all directions to spread the pastry into a circle about 1 inch large than the top of your pie plate. (To prevent wax paper from slipping, sprinkle a drop or two of water on the counter or work area before placing the dough down.) Repeat with each circle and set aside.
Peel top sheet of waxed paper from one of the pastry circles, but leave the back on. Then, rest dough on hand and wrist and gently lift onto a 9-inch pie plate and ease dough into place inverting it, but do not stretch. Set pie plate and dough for top aside.
For the filling, assembly, and baking:
Preheat oven to 375° F. Core apples and slice into 1/8-inch half-moons. Add to a large bowl, then toss with cinnamon, tapioca, sugar, flour, and lemon.
Add an even layer of flour to the pie plate assembled in Step 4 of making the dough (this will soak up some of the moisture and ensure a crisp bottom crust). Being careful not to add too much of the moisture pooled at the bottom of the bowl, transfer the filling into the prepared crust.
Very carefully peel wax paper off of the second pastry circle and invert over the filling. Trim the edges, if there's excess, and flute the edges if able. Brush with whisked egg white.
Place the pie onto a rimmed baking sheet, and create a bag out of parchment paper. This can either be done by taking two pieces of parchment paper, large enough to fit the entire pie inside, and stapling three of the edges together—use plenty of staples here, the idea is that steam cannot escape. Then carefully put the pie inside, staple it in, and place onto the baking sheet. This works great for presentation—as you can serve the pie with the bag. Alternatively, pull a sheet of parchment across the pie and tuck in the edges, then pull another sheet to layer over the first perpendicularly and tuck in the edges carefully so that the pie is pinning it down.
Place the pie in the oven and bake for 50 minutes. After 50 minutes, remove it from the oven and raise the temperature to 425° F. Using a pair of scissors, carefully (steam will escape and the pie will be hot!) cut open a hole that goes just up to the edges of the pie, roughly 8 inches in diameter (so that it leaves the top of the pie bare, but still covers the edges). Once the oven has reach temperature, return it for 8 minutes, or until golden brown. Remove and eat (with ice cream!).