Prepare the Flaky Pie Dough: Line a small metal or plastic bowl with aluminum foil. Place 6 tablespoons butter in a small heavy saucepan; cook over medium, stirring constantly, until butter begins to turn golden brown, 6 to 8 minutes. Remove from heat; immediately pour butter into prepared bowl. Freeze until butter has solidified, about 1 hour. Lift foil from bowl; remove foil from butter, and cut into small pieces.

Stir together flours and salt in a medium bowl. Sprinkle browned butter pieces and cold butter pieces over flour mixture. (Do not mix into flour.) Place bowl in freezer 10 minutes.

Remove from freezer. Using your hands, pinch and smear butter pieces into the flour mixture to evenly distribute. (Return bowl to freezer for 10 minutes if butters begin to soften.)

Add 1⁄2 cup cold buttermilk, and stir just until dough begins to come together, stirring in more buttermilk, 1 tablespoon at a time, if needed. Dough should be dry and formed into a mass; pieces of butter should be visible.

Divide dough in half. Press each half into a round, flat disk; wrap in plastic wrap, and refrigerate until thoroughly chilled, at least 1 hour.

Remove 1 dough disk from refrigerator and unwrap. Roll dough into a 13-inch circle on a lightly floured surface. Fit dough into a 9-inch pie plate, making sure dough is evenly draped with no stretching or pockets. Place pie plate in refrigerator.

Remove remaining dough disk from refrigerator and unwrap. Roll dough to 1⁄8-inch thickness on lightly floured parchment paper. Cut into about 8 (1-inch-wide) strips. Transfer dough strips, with parchment paper, onto a large plate or baking sheet, and chill until ready.

Prepare the Filling: Stir together apple slices, dried apples, butter, and apple butter in a large skillet. Cook apple mixture over medium-high, stirring often, until apples are tender-crisp, 10 to 15 minutes. Remove from heat. Add lemon juice, orange zest, grated ginger, and vanilla. Gently stir to combine. Sprinkle cinnamon, nutmeg, cloves, and cardamom over apple mixture, and gently stir to combine. Sprinkle with brown sugar, flour, and salt, and gently stir to combine.

Spoon Filling into chilled pie shell. Arrange dough strips in a lattice pattern on top. Trim excess dough, and discard. Press edges with a fork or crimp by pinching with your fingers to make a decorative pattern. Freeze 20 minutes.

Preheat oven to 425ºF. Brush crust with beaten egg, and sprinkle lightly with demerara sugar. Bake in preheated oven 15 minutes; reduce oven temperature to 350°F, and bake until Filling is bubbly and crust is golden brown, about 45 more minutes. Cover pie with aluminum foil after 30 minutes to prevent overbrowning. Let cool completely (4 to 6 hours) before slicing.