In a food processor, combine the flour, 3/4 teaspoon of salt, and 1 tablespoon of granulated sugar; process to blend thoroughly.

Add the 8 tablespoons of chilled butter and shortening pieces and pulse about 12 to 15 times, or until the mixture resembles crumbs and there are still some small pea-size pieces of fat visible.

Alternatively, use your fingers or a pastry blender to incorporate the fat into the flour mixture. If using a food processor, transfer the mixture to a large bowl.

Sprinkle 3 tablespoons of buttermilk over the mixture and toss to moisten. Add more buttermilk, about 1 teaspoon at a time, tossing and mixing gently until dough begins to clump.

Pour the dough mixture out onto a floured surface and knead a few times, moistening with small amounts of buttermilk if necessary to form the dough. Don't over-moisten or overwork the dough.

Shape the dough into two uniform flat disks and wrap each in plastic wrap. Chill for about 30 minutes.

On a floured surface with a floured rolling pin, begin to roll out the first disk. As you roll, keep moving the dough gently and add more flour to the surface as needed. Roll from the center a few times in one direction, rotate the dough about a quarter turn, and roll in that direction a few times. Keep rolling and turning until you have a circle about 12 inches in diameter. Drape the dough gently over your floured rolling pin and unfold it over the pie plate.

Press the dough over the bottom and up the sides of the pie plate. Leave the overhang until you're ready to crimp the edge.

In a saucepan over low heat, combine the corn syrup, 3 tablespoons of butter, brown sugar, cornstarch, vanilla, the 1 teaspoon of cinnamon, and 1/4 teaspoon of salt. Heat until hot and butter is melted. Set aside to cool slightly.

Prepare the apples and toss with lemon juice in a large bowl. Add the syrup mixture to the apples and then spoon them into the prepared bottom crust.

Roll the second chilled disk of dough out as the first. Transfer to the filled pie pan and trim the overhang to about 1 inch. Fold the overhang under, creating a ropelike edge all around the pie; crimp as desired.

Heat the oven to 375 F.

Place a sheet pan or baking sheet on the oven rack.

Combine the egg yolk and milk; blend well. Lightly brush over the top crust. Cut several slits in the top crust to vent.

Place the pie on the hot sheet pan or baking sheet.

Bake the pie for 50 to 60 minutes, checking for over-browning at about 40 minutes. Fit the pie with a pie shield or make a foil ring to fit over the edge, if necessary.

When the crust is nicely browned and apples are tender, remove the pie from the oven and let it cool completely on a rack.