In a large bowl, whisk together the all-purpose flour, rye flour, sugar, and salt. Add the cold, cubed butter and use a pastry cutter or your fingertips to work it into the flour mixture until it resembles coarse meal with some pea-sized pieces remaining. Toss in the grated cheddar cheese, ensuring it's evenly distributed.
Sprinkle 6 tablespoons of ice water over the mixture and stir with a fork until the dough begins to come together. If needed, add more water, 1 tablespoon at a time. The dough should hold together when squeezed but shouldn't be sticky.
Divide the dough in half, form each portion into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes.
Peel, core, and slice the apples into ¼-inch thick slices. In a large bowl, toss the apples with lemon juice to prevent browning. Add the sugar, flour, cinnamon, nutmeg, allspice, and salt, tossing until apples are evenly coated.
Preheat your oven to 425°F (220°C) with a rack in the lower third position.
On a lightly floured surface, roll out one disk of dough to a 12-inch round, about ⅛-inch thick. Carefully transfer to a 9-inch pie plate, gently pressing into the bottom and sides. Trim excess dough, leaving a 1-inch overhang.
Pour the apple filling into the crust, mounding slightly in the center. Dot the filling with the small pieces of butter.
Roll out the second disk of dough and either place it whole over the filling or create a decorative lattice pattern. If using a whole top crust, cut a few slits to allow steam to escape.
Trim excess dough from the top crust and fold the overhang from the bottom crust over the edges of the top crust. Crimp the edges decoratively to seal. Brush the top with egg wash and sprinkle with turbinado sugar and additional grated cheddar (if using).
Place the pie on a baking sheet (to catch any drips) and bake for 20 minutes at 425°F. Reduce the temperature to 375°F (190°C) and continue baking for 35-40 minutes more, until the crust is golden brown and the filling is bubbling.
If the edges are browning too quickly, cover them with aluminum foil or a pie shield.
Allow the pie to cool on a wire rack for at least 2 hours before serving. This resting period is essential for the filling to set properly.