8 tablespoons unsalted butter, cut into small cubes
1 large egg yolk
10 to 12 tablespoons ice water
5 cups thinly sliced peeled tart apples
3/4 cup plus 2 tablespoons apple juice, divided
3/4 cup sugar
3/4 teaspoon ground cinnamon
1/4 teaspoon salt
1/4 teaspoon ground nutmeg
3 tablespoons cornstarch
2 cups fresh cranberries
1 large egg, lightly beaten
1 tablespoon 2% milk
In a large bowl, combine all-purpose flour, cake flour, sugar, salt and baking powder; cover and freeze 30 minutes. Place lard and butter in a separate bowl; freeze 30 minutes. Place flour mixture in a food processor; pulse until blended. Add lard and butter; pulse until mixture is the size of peas.
Transfer flour mixture to a large bowl. Mix together egg yolk and 10 tablespoons ice water; gradually add to flour mixture. Toss with a fork until dough holds together when pressed, adding more water if needed. Divide dough into thirds. Shape each into a disk; wrap in plastic wrap. Refrigerate 1 hour or overnight.
In a large saucepan, combine apples, 3/4 cup apple juice, sugar, cinnamon, salt and nutmeg; bring to a boil over medium heat, stirring occasionally. Combine cornstarch and remaining 2 tablespoons juice until smooth; add to saucepan. Return to a boil, stirring constantly. Cook and stir 1 minute or until thickened. Remove from the heat; cool to room temperature. Stir in cranberries.
On a lightly floured surface, roll 1 dough portion into a 1/8-in.-thick circle; transfer to a 9-in. pie plate. Trim crust even with rim. Pour fruit filling into crust.
Roll out another portion of dough; make a lattice crust. Place over filling. Trim and seal edge. Roll out remaining portion of dough to a 14x4-in. rectangle. Cut four 14x1/4-in. strips. Place 2 strips side by side; twist together. Repeat with remaining 2 strips.
For decorative cutouts, cut remaining dough out with a 1-1/2-in. leaf-shaped cookie cutter. With a sharp knife, lightly score cutouts to resemble veins on leaves. Combine egg and milk; lightly brush lattice and edge of crust with egg mixture. Place twists along edge of pie crust, pressing lightly to adhere. Overlap cutouts on the lattice and along edge of pie. Refrigerate pie until crust is well chilled, about 30 minutes.
Preheat oven to 400°. Brush cutouts with egg mixture. Cover edges loosely with foil coated with cooking spray. Bake 20 minutes. Remove foil; bake until crust is golden brown and filling is bubbly, 30-40 minutes longer. Cool on a wire rack.