Dough

1. Combine flour, salt, and sugar in a large mixing bowl. Add grated cheese and toss to fully coat in flour. Cube butter into roughly ½-inch pieces and toss with dry ingredients. Cut in cold butter with a pastry blender, breaking large pieces into smaller, pebble-sized pieces. Use your fingers to “snap” the butter, flattening it into large flakes. Continue coating butter in flour until mixture is well-blended. It should look like parmesan cheese with a few pebble-sized pieces throughout. 
2. Place the apple cider vinegar and a few ice cubes in a measuring cup, then pour in water to make a ½ cup of total liquid.
3. Make a well in the center of the flour mixture and add ¼-cup ice water mixture. Toss thoroughly. Continue adding water a little bit at a time. Scoop the mixture from the bottom of bowl on top of the dough mound and press gently. Continue scooping and pressing until the dough comes together in a solid mass. Turn out onto a lightly floured surface and shape into a fat log. Cut log in half and shape each half into a disc, wrap tightly in plastic, and refrigerate for at least one hour, or overnight. 
4. Once dough has chilled, roll out on a lightly floured surface. Line the bottom of a 9 Inch Pie Pan, leaving a 2-inch overhang. Chill in the refrigerator while you roll the second disc of dough. Roll the second disc of dough into a circle at least two inches larger than the diameter of the pie pan. Cut six wide strips of dough. This will be your lattice. Place on a parchment-lined sheet pan and refrigerate while you arrange the apples inside the bottom crust. 

Filling

1. Toss sliced apples with lemon juice. Combine remaining ingredients and toss with the apples. Working from the outside in, layer apples tightly in the pie pan. Continue layering and stacking until apples reach the top of the pie crust. Mound remaining apples in the center. 
2. Place three strips of the wide lattice vertically on top of the pie. Pull the middle strip down and lay one horizontal strip across. Pull the two outer strips down and lay the next horizontal strip across the top. Repeat the process with outer strips and lay the final horizontal strip across the bottom of the pie. Trim all lattice pieces just to the edge of the pie plate. With the overhang of the bottom crust, roll crust on top of the lip of the pan to cover the raw edge of lattice pieces. Continue rolling to create a crust. Crimp as desired, or use the tines of a fork to create a decorative edge.
3. Chill assembled pie for 15 minutes. Prepare egg wash by beating yolk with heavy cream. 
4. Preheat oven to 425 degrees F. Place the chilled pie on a sheet pan lined with parchment paper and use a pastry brush to apply the egg wash to the pie. Sprinkle the top with demerara sugar. Bake in lower half of the oven for 25 minutes, until the pastry sets and begins to turn golden. Lower the temperature to 375 degrees and bake for 30 more minutes, until the pastry is golden brown (no pale pies!) and the filling bubbles through the lattice. 
5. Cool completely before serving.