Make the Crust: Place the flour, salt and sugar in a food processor and pulse a few times to combine. Scatter the pieces of butter over the flour mixture, using your fingers to toss and coat the butter with some flour. Cut the butter into the flour mixture with five 1-second pulses. Add the shortening and continue to pulse until the flour is ple yellow and resembles coarse cornmeal with butter bits no larger than small peas, about four more 1-second pulses. Turn the mixture into a medium bowl.
Sprinkle all but 1 tablespoon of the ice water over the mixture. Use a rubber spatula to fold the water into the flour mixture. Press down on the dough with the spatula until the dough sticks together, adding up to 1 tablespoon of remaining ice water if the dough does not come together. Divide the dough into two balls, one slightly larger than the other. Flatten each into disks, dust lightly with flour, wrap separately in plastic wrap and refrigerate for at least 30 minutes.
Remove dough from the refrigerator; let stand at room temperature to soften slightly, about 10 minutes. Preheat oven to 400 degrees.
Make the Filling: Toss the fruit with sugar, orange zest, lemon juice, vanilla extract and tapioca; let stand for 15 minutes.
Assemble the Pie: Roll the larger dough disk on a lightly floured surface into a 12-inch circle, about 1/8-inch thick. Transfer and fit dough into 9-inch pie plate, leaving dough that overhangs the lip in place. Turn the fruit mixture, including juices, into the pie shell. Scatter the butter pieces over the fruit. Refrigerate until ready to top with the remaining dough.
Roll the smaller disk on a lightly floured surface into a 10-inch circle. Lay over the fruit. Trim the top and bottom dough edges to ½-inch beyond the lip of the pie plate. Tuck this rim of dough underneath itself so that the folded edge is flush with the edge of the pie plate. Flute the dough or press with a fork to seal. Cut four slits at right angles on the top of the dough to allow steam to escape. Place in the freezer for 10 minutes before baking.
Place the pie on a baking sheet; bake until the top crust is golden, 20 to 25 minutes. Reduce the oven temperature to 350 degrees and continue to bake until the juices bubble and the crust is golden brown, 30 to 40 minutes longer.
Transfer the pie to a wire rack; let cool for at least 2 hours before serving so the juices have time to thicken. Leftovers can be stored at room temperature, covered, for up to 3 days.