For Pie Crust:

Beat the butter and shortening in the large bowl of an electric mixer until blended. Transfer the mixture to a piece of plastic wrap and form into a flat disc about 1 inch thick. Freeze firm.

Put the flour and salt in a large bowl. Grate the frozen butter mixture. Working quickly, and using your fingertips rub the butter and flour together until the mixture looks mealy and the flour is coated with shortening. Add the water, stirring constantly, until a dough forms that holds together. Add Additional water by tablespoonfuls if necessary. Turn the dough out onto a lightly floured rolling surface. With the heel of your hand push the dough down and forward against the rolling surface about 6 times until the dough looks smooth. Form into 2 disks, wrap in plastic wrap and refrigerate for atleast 30 minutes.

For Apple Filling:

Preheat the oven to 475° F.

In a large bowl mix the apple slices and lemon juice together. In a small bowl mix the salt, brown sugar, granulated sugar, cinnamon and nutmeg together. Mix half of the sugar mixture into the apples. Stir in the melted butter. Spread in a single thickness on a baking sheet with 1-inch high sides and bake for about 18 minutes until the juices caramelize and most are evaporated, stirring once. Cool thoroughly. Mix the remaining sugar mixture, tapioca and cornstarch into the apples.

Reduce the oven temperature to 375° F.

On a lightly floured surface, roll one disk of pie dough into a circle about 4 inches wider than the bottom of the pie pan. Press the dough into the pie pan. Trim the edges. Put the cooled apple filling in the crust. Roll out the second disk of dough and drape it over the apples. Press the bottom and top dough together firmly to seal. Trim to 1/2 - inch overhang and flute the edges.

Brush the egg wash and bake for about 50 minutes until the crust is golden brown.

Increase the oven temperature to 450 ° F. Brush the top crust with the sugar syrup. Bake until the syrup caramelizes, about 7 minutes.