Preheat the oven to 400 degrees F. For the pie crust: In a food processor add the flour, sugar, and salt. Pulse to combine. Then cut the cold butter and shortening into cubes and place in the food processor. Pulse again until the mixture resembles small peas.
Add 2/3 cup ice cold amaretto and pulse again. If the dough looks too dry add 1-2 more tablespoons and pulse. Dump the dough out onto a piece of plastic wrap. Press into a rectangle shape and wrap well. Refrigerate for at least 30 minutes.
Meanwhile, peel and slice the apple into thin wedges. (You can do this in the food processor as well, using the slicer attachment.) Place the apples in a large bowl and drizzle lemon juice over the top. Toss to coat. Add the sugar, flour, vanilla, and apple pie spice. Toss to coat again.
Take the dough out of the refrigerator. Cut 1/3 of the dough off one end. Wrap the smaller piece and place back in the fridge. Place the larger section on a well-floured piece of wax paper. Roll into a large 12 X 17 inch rectangle. Carefully flip the dough over onto a 10 X 15 inch jelly roll pan and peel off the wax paper. Roll the edges under and crimp with a fork.
Spread the apple mixture over the crust. Roll out the remaining piece of crust to 1/8 inch thick and use a decorative 2-inch cookie cutter to cut 24 pieces of dough. Lay the shapes evenly over the surface of the apples.
Whisk the egg with 1 tablespoon water. Brush the egg-wash over the edges of the crust and the dough shapes. Bake for 30-40 minutes, until gold and bubbly. If the crust is getting dark before the apples are fully cooked, lay a sheet of foil over the top of the pie.
Once the pie is out of the oven, drizzle the caramel sauce over the top. Use as much as you like! Allow the pie to cool for at least 10-15 minutes before cutting.