Combine 3/4 cup sugar, cinnamon, and 2 teaspoons tapioca in a large bowl, stirring well. Add apple slices; toss well. Set aside.
Combine 3/4 cup sugar and 3/4 cup flour in a bowl, stirring well. Cut in butter with a pastry blender until crumbly. Set aside.
Combine 1 cup flour, shortening, and salt with a pastry blender until crumbly. Sprinkle ice water, 1 tablespoon at a time, evenly over surface; stir with a fork just until dry ingredients are moistened. Shape pastry into a ball.
Roll pastry to 1/8-inch thickness on a lightly floured surface. Fit into a 9-inch pieplate; trim off excess pastry along edges. Fold edges under, and crimp. Sprinkle with remaining 2 teaspoons tapioca. Spoon apple mixture into crust, mounding in center. Sprinkle with butter mixture, covering apple mixture completely. Place on a baking sheet lined with aluminum foil.
Bake at 400° for 25 minutes. Shield pie from overbrowning by placing a square of aluminum foil lightly on pie. Bake 20 more minutes. Cool on a wire rack.
Tips: The shortening-flour mixture for the pastry should resemble small peas, not cornmeal, for maximum flakiness. Be careful not to work the pastry mixture for the piecrust anymore than absolutely necessary to "gather" it together into a ball or it will be dense, tough, and lose its flakiness. If your shortening is not cool, you can chill the shortening mixture before the water is added.