Mix flour, salt, and sugar in food processor fitted with steel blade. Scatter butter pieces over flour mixture, tossing to coat butter with a little flour. Cut butter into flour with five 1-second pulses. Add shortening and continue to cut it in until flour is pale yellow and resembles coarse cornmeal with butter bits no larger than small peas, about four more 1-second pulses. Turn mixture into medium bowl.
Sprinkle all but 1 tablespoon of the ice water over mixture. With blade of rubber spatula, use folding motion to mix. Press down on dough with broad side of spatula until dough sticks together, adding up to 1 tablespoon of remaining ice water if dough does not come together. Divide dough into two balls, one slightly larger than the other. Flatten each into 4-inch-wide disk. Dust lightly with flour, wrap separately in plastic, and refrigerate at least 30 minutes.
Remove dough from refrigerator; let stand at room temperature to soften slightly, about 10 minutes. Heat oven to 400 degrees. Toss fruit with sugar, lemon juice and zest, spices, almond extract, brandy, and tapioca; let stand for 15 minutes.
Roll larger dough disk on lightly floured surface into 12-inch circle, about 1/8-inch thick. Transfer and fit dough into 9-inch Pyrex pie pan, leaving dough that overhangs the lip in place. Turn fruit mixture, including juices, into pie shell. Scatter butter pieces over fruit. Refrigerate until ready to top with remaining dough.
Roll smaller disk on lightly floured surface into 10-inch circle. Lay over fruit. Trim top and bottom dough edges to 1/2-inch beyond pan lip. Tuck this rim of dough underneath itself so that folded edge is flush with pan lip. Flute dough in your own fashion, or press with fork tines to seal. Cut four slits at right angles on dough top to allow steam to escape. If pie dough is very soft, place in freezer for 10 minutes before baking.
Place pie on baking sheet; bake until top crust is golden, 20 to 25 minutes. Reduce oven temperature to 350 degrees and continue to bake until juices bubble and crust is golden brown, 30 to 40 minutes longer.
Transfer pie to wire rack; let cool to almost room temperature so juices have time to thicken, from 1 to 2 hours.