Blend together flour, butter, shortening, and salt in a bowl with your fingertips (or pulse in a food processor) until mixture resembles coarse meal with some roughly pea-size butter lumps. Drizzle evenly with 5 tablespoons ice water and gently stir with a fork (or pulse) until incorporated.
Squeeze a small handful: If it doesn't hold together, add more ice water, 1 tablespoon at a time, stirring (or pulsing) until just combined. (Do not overwork mixture, or pastry will be tough.)
Turn out dough onto a work surface and divide into 6 portions. With the heel of your hand, smear each portion once or twice in a forward motion to help distribute fat. Gather all of dough together with scraper and press into 2 balls, then flatten each into a 5" disk. Chill dough, each disk wrapped tightly in plastic wrap, until firm, at least 1 hour.
Toss nectarines, plums, raspberries, lemon juice, and 1/4 cup of sugar in a large bowl.
Grind tapioca to a powder in grinder, then whisk together with cornstarch, salt, and remaining 1/2 cup sugar in a small bowl (do not toss with fruit until dough is rolled out).
Roll out 1 disk of dough on a lightly floured surface with a lightly floured rolling pin, into a 13"-14" round, then fit into a 9" pie plate and trim edge, leaving a 1" overhang.
Roll out remaining disk to a 14" round. Slide round onto a baking sheet. Cover and chill both crusts for 1 hour.
Preheat oven to 400° with rack in lower third. Toss sugar mixture with fruit and transfer to chilled pie shell.
Using fluted pastry wheel (or other utensil), cut chilled dough round into 3/4"-wide strips. Weave strips over filling, forming a lattice. Trim strips 1/2" shorter than dough overhang. Fold overhang over strips and crimp edges decoratively.
Brush beaten egg over crust edges and lattice. Sprinkle 1 tablespoon sugar over crust. Set pie on a large rimmed baking sheet lined with foil.
Bake pie 30 minutes. Reduce oven temperature to 350° and continue to bake, checking frequently and covering edge with a strip of foil or pie shield if crusts are browning too fast, until crusts are golden brown and filling is bubbling, 40-50 minutes more.
Cool pie to room temperature on a rack, at least 2 hours.
