Heat oven to 400° F.
On a lightly floured surface, roll half of the pastry dough into a round about 13-inches in diameter. Lower dough carefully into a deep 9-inch pie dish, pressing it into place gently with your fingers. Trim excess dough, leaving a 1-inch overhang all around. Place dish into the refrigerator to chill as you prepare the filling.
In a large bowl, gently toss the plums, blackberries, sugar, vanilla, lemon juice, lemon zest, and tapioca. Spoon the fruit mixture into the chilled pie plate.
Roll the remaining disc of dough into a circle. If you would like to make a lattice top, slice the rolled dough into 1-1½-inch strips and weave it into place on the surface of the pie. Or, keep it simple and just drape the rolled round over the filling, trim the edges, and crimp them to seal. Slash several vents into the top of the crust with a sharp knife. Brush dough all over with beaten egg and sprinkle generously with turbinado sugar.
Bake until golden, about 20 minutes, then lower heat to 375° F and continue to bake until juices are bubbling, about 40 minutes more. If crust darkens too deeply, tent with tin foil. Remove from oven and let cool at least 2 hours before serving.