How to Make It
In a food processor, combine the flour, salt, shortening, and ½ cup of the butter until the mixture forms pea-size crumbs. Break the egg into a measuring cup and beat lightly; add the vinegar and enough cold water to measure ½ cup. Slowly add the egg mixture to the flour mixture, pulsing until a soft dough forms.
Divide the dough in half. Shape each half into a flat disk. Wrap in plastic and refrigerate for at least 1 hour.
On a lightly floured surface, roll out 1 disk into a 14-inch circle. Place in a 2-inch-deep 9-inch pie plate.
In a medium bowl, gently toss the peaches, strawberries, tapioca, lemon juice, and ½ cup of the sugar. Spoon the fruit mixture into the pie plate and dot with the remaining 4 tablespoons of butter; refrigerate.
Roll out the second disk of dough into a 14-inch circle. Drape it over the pie and trim the edge to a ½-inch overhang. Fold under, pressing to seal. Crimp the edge, if desired. Refrigerate for at least 30 minutes.
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, 18 to 22 minutes. Reduce heat to 375° F and continue to bake until the juices bubble, 30 to 40 minutes more.
Transfer to a rack and cool for 2 hours before serving.