In a large bowl, whisk together 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 crumbs with some pea-sized pieces remaining.
Gradually add ice water, 1 tablespoon at a time, stirring with a fork until the dough begins to come together.
Divide the dough into two equal portions, shape into disks, wrap in plastic, and refrigerate for at least 1 hour or up to 2 days.
In a large bowl, combine sliced apples and cranberries.
In a separate bowl, mix together granulated sugar, brown sugar, cornstarch, lemon juice, cinnamon, nutmeg, allspice, and vanilla extract.
Pour the sugar mixture over the fruits and toss until evenly coated. Let stand for 15 minutes to allow juices to develop.
On a lightly floured surface, roll out one disk of dough into a 12-inch circle, about ⅛-inch thick.
Carefully transfer to a 9-inch pie dish, pressing gently into the bottom and sides. Trim the edges, leaving a 1-inch overhang.
Roll out the second disk to the same thickness and use a pizza cutter or knife to cut into 10-12 strips of even width (about ¾-inch wide) for the lattice.
Pour the fruit filling into the prepared bottom crust, spreading evenly. Dot the top with small pieces of butter.
Arrange half of the dough strips in parallel rows across the pie, spacing them evenly.
Fold back every other strip halfway, then place a new strip perpendicular to them. Unfold the folded strips over the new strip.
Fold back the alternate strips and place another perpendicular strip. Continue this weaving process until the lattice is complete.
Trim excess dough from the strips and fold the overhang from the bottom crust over the edges of the lattice strips. Crimp or flute the edges as desired.
Preheat oven to 425°F (220°C) and place a baking sheet on the lower rack.
Brush the lattice and edges with beaten egg and sprinkle with turbinado sugar.
Place the pie on the preheated baking sheet and bake for 20 minutes.
Reduce the oven 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 through the lattice.
If the edges start to brown too quickly, cover them with aluminum foil or a pie shield.
Allow the pie to cool completely on a wire rack for at least 3 hours before slicing to let the filling set.