To make the crust, combine three cups flour, one tablespoon sugar and one teaspoon salt in a medium bowl. Use a pastry cutter to cut in 3/4 cup butter and shortening until the pieces become the size of peas. Add in 6 tablespoons of water and mix gently with a rubber spatula or wooden spoon until the dough just starts to come together. Add more water one tablespoon at a time if needed. The dough should not be sticky, and you should be able to see the pieces of butter and shortening flecked throughout. Do. Not. Over. Mix.
Once the dough comes together, split it into two balls, then gently shape them into discs, wrap and refrigerate for at last 30 minutes (or up to two days).
While the dough is chilling, make the filling. Peel, core and slice the apples (not too thin) and place in a large mixing bowl. Add the remaining 1/4 cup flour, 1/2 cup sugar, 1/2 teaspoon salt, and cinnamon. Toss the apples until evenly coated.
It's important that the dough stay cold while you roll it out, so keep it in the refrigerator until just ready. Prepare the counter and rolling pin by dusting each liberally with flour. Remove one disc of dough from the refrigerator and roll to about 12 inches around, rotating the disc as you go to keep it from sticking to the counter. Use a pizza cutter or knife to trim the edges so that the shape is as round as possible.
Transfer the dough to a pie plate, making sure it's distributed evenly around the sides. Fill the shell with the apples and dot the top with the remaining two tablespoons of butter. Use your fingers or a pastry brush to lightly wet the perimeter of exposed dough around the edges.
Roll out the second disc of dough and place it over top of the apples. Use your fingers to go around and press the top crust to seal with the bottom crust, then neatly tuck it all under. Use your thumb and index finger on each hand to push and pull the crust to form a fluted edge.
Transfer the pie to a freezer for at least one hour, or wrapped tightly, for up to 6 months.
When ready to bake, preheat the oven to 350 degrees. Use a sharp knife to slit three holes in the top of the pie (be creative, you artists!). Then use a pastry brush to paint the entire thing with the beaten egg mixture, being careful not to close up the vents you just created.
Transfer the pie to the oven and allow it to bake for about 90 minutes, rotating half way through to ensure even cooking. The crust should be golden brown and the filling oozing around the vents. If the crust begins to get too brown, lower the oven to 325 and tent the top with a piece of foil.
Allow the pie to rest for at least one hour before cutting into it, otherwise the filling will ooze all over. Serve with a dollop of whipped cream or vanilla ice cream. Swoon.