Make the dough in a food processor by combining half of the flour, sugar, and salt together and pulsing until combined. Add in the butter and shortening pieces and process for about 15 to 20 seconds until combined. Add in the remaining half of the flour and pulse until all the flour is incorporated. Sprinkle in cold water and vodka and pulse 5 or 6 times until combined into a dough. Divide into two even pieces and wrap in plastic wrap. Refrigerate 1 hour.
In a large mixing bowl, toss together peaches and 1 cup sugar. Let sit 1 hour. Scoop out 1/4 cup of the juices and set aside. Drain the rest of the excess liquid from the peaches using a colander. Return the peaches to the mixing bowl , add in reserved juices, and toss with cornstarch, lemon juice, cinnamon, nutmeg and salt.
Roll out your bottom crust into a 12-inch circle. Make sure you use a well floured surface and rolling pin to prevent sticking. Roll the circular dough loosely around the rolling pin to transfer to a 9-inch pie dish (deep dish). Be gentle and press it carefully into the plate letting the edges hang over. Refrigerate until ready to use.
Prepare your lattice crust by rolling the remaining dough half into a 13 by 10 inch rectangle. Transfer to a baking sheet lined with parchment paper. Cut into 8 even strips. Freeze on the baking sheet while oven preheats.
Preheat oven to 425 degrees.
Place the prepared pie plate on a baking sheet (in case of spillage). Scoop the peach filling into the pie shell. Lay 4 of the lattice strips on top of the peaches, giving even spacing between. Weave the remaining lattice pieces perpendicular, lifting every other piece to create a lattice. Trim the excess dough that is hanging around the pie plate with kitchen scissors or a sharp knife. Pinch the lattice and crust edges together and then crimp the dough evenly around the pie using your fingers.
Brush the crust with the beaten egg and sprinkle with 1 TB sugar. Bake in the preheated oven for 25 minutes. Reduce the temperature to 375 degrees and cook 30 to 35 minutes more until crust is golden brown and juices are bubbling and set. Transfer pie dish to a wire rack to cool.