Apple, Pear, and Cranberry Pie

For the filling, use lightly firm pears and crunchy, tart apples, and before serving, let this lush pie cool and infuse for at least two hours.
Prepare the cake: preheat the oven to 375 degrees. Combine the flour, the next 2 ingredients, and 2 tablespoons of sugar in a medium bowl. Use a blender or fork to whisk together the butter and ground flour until the mixture resembles small peas. Pour the flour mixture on 1 side of the bowl. Drizzle 1 tablespoon of cider along the edge of the flour mixture. Gently pour a small amount of the flour mixture into the cider until it is moistened; transfer the moistened mixture to the opposite side of the bowl. Repeat the process with the remaining 7 tablespoons of the cider and flour mixture, adding up to 2 tablespoons of cider, 1 tablespoon bake in turn until a shortbread is formed. Form 2 circles from the dough; wrap in plastic wrap. Refrigerate for 1 hour.Roll out 1 circle of dough into a 13-inch circle on a lightly floured surface. Place the dough in a 9-inch glass pie dish; the size of the dough should be 2 inches larger than the size of the pie dish. Mix the egg and 1 teaspoon of water. Combine 1/8 tsp cinnamon and remaining 2 tsp sugar. Set the mixes aside.Prepare the filling: Mix the apples with the following 5 ingredients; let stand for 15 minutes. Add the cranberries; spoon them into the cake and place them in a mound in the center. Brush with 1 tablespoon of butter.Roll out the remaining dough circle into a 13-inch circle on a lightly floured surface. Place the dough on top of the filling. Bend the edges; crimp to seal.Brush the top with egg mixture and sprinkle with sugar and cinnamon. Place the cake on a baking sheet lined with aluminum foil; make 4-5 slots in the top cake.Bake at 375 degrees for 40 minutes. Cover loosely with foil and bake for another 50 minutes. Cool on a wire rack 2 hours before slicing.