In a large bowl, mix the flour and salt. Add the cold butter cubes and shortening. Cut them into the flour using a pastry cutter or fork until the mixture resembles coarse crumbs.

Gradually add ice water, 1 tablespoon at a time, mixing with a fork until the dough begins to clump together.

Divide the dough into two equal portions and shape each into a disk. Wrap in plastic wrap and refrigerate for at least 1 hour or until firm.

Peel, core, and thinly slice the apples. Toss them in a large bowl with lemon juice to prevent browning.

In a separate bowl, mix the white sugar, brown sugar, flour, cinnamon, nutmeg, and salt.

Add the sugar mixture to the apples and toss until evenly coated.

Preheat your oven to 400°F (200°C). Place a baking sheet on the oven rack to catch any drips.

On a lightly floured surface, using a rolling pin, roll out one disk of dough into a circle large enough to line a 9-inch pie plate. Transfer the dough to the pie plate, leaving excess dough hanging over the edge.

Fill the pie shell with the apple mixture, mounding slightly in the center. Dot with the tablespoon of butter.

Roll out the second disk of dough and place it over the filling. Trim excess dough, leaving about a 1-inch overhang. Fold the top crust under the bottom crust edge, then crimp the edges to seal. Cut slits in the top crust to allow steam to escape.

Brush the top crust with beaten egg and sprinkle with sugar.

Cover the edges of the pie with aluminum foil to prevent over-browning. Place the pie on the preheated baking sheet in the oven.

Bake at 400°F (200°C) for 20 minutes, then reduce the oven temperature to 350°F (175°C) and bake for an additional 35-40 minutes or until the crust is golden brown and the filling is bubbly.

Remove the pie from the oven and cool on a wire rack for at least 2 hours before serving. Serve warm or at room temperature with a scoop of vanilla ice cream for a classic dessert experience.