In a food processor (or in a large bowl, using a pastry blender, 2 crossed knives, or your fingertips), combine the flour, salt, 1/2 cup butter, and shortening until the mixture is crumbly and the size of peas. Break the egg into a measuring cup and beat lightly; add the vinegar and enough cold water to measure 1/2 cup. Slowly add the egg mixture to the flour mixture and stir until a soft dough forms. Divide in half. Shape each half into a flat disk and wrap in plastic. Refrigerate for at least 1 hour.
Summer Fruit Pie:
Lightly flour the work surface. Roll 1 disk of the dough into a circle approximately 14 inches in diameter and 1/8 inch thick. Place the dough in a 9-inch pie plate 2 inches deep, cover, and refrigerate. In a medium bowl, gently toss together the peaches, strawberries, tapioca, lemon juice, and 3/4 cup of the sugar. Spoon the fruit mixture into the pie plate, sprinkle with 4 tablespoons butter, and refrigerate. Roll out the second disk of dough, drape it over the pie, and press the edges together to seal. Refrigerate for at least 30 minutes and up to 4 hours. Heat oven to 425° F. Lightly brush the top crust with cold water and sprinkle with the remaining sugar. Cut several slashes into the top crust. Bake until golden, about 20 minutes. Reduce heat to 375° F and continue to bake until the juices are bubbling, 30 to 40 minutes more. Transfer to a rack for at least 3 hours to cool and thicken before serving.