Combine all dough ingredients except the ice water in a large bowl. With clean hands, blend the mixture together until it looks like coarse meal with some lumps in it.
Sprinkle ice water over mixture and stir lightly with a fork. Squeeze a handful of dough together. If it doesn?t keep together, add a bit more water.
Divide the dough in half and make two chubby disks about 5 inches across. Wrap the disks separately in plastic wrap and chill for about an hour. Once chilled, take out one disk and put it on a well floured board. Sprinkle some flour onto the top of the disk. Thump the disk with your rolling pin several times. Turn it over and thump the other side. Sprinkle more flour onto the top of the crust if needed to keep the pin from sticking and roll the crust out from the center in all directions.
When it is an inch or so larger than your pie pan, fold the dough over the top of the pin and lay it in the pie pan carefully.
Combine the peaches, strawberries, sugar, nutmeg, salt, flour, and tapioca in a big bowl and gently mix all together. Spoon the filling into pastry in the pie plate (with the dough) and dot with little pieces of butter.
To make a lattice, once the dough is rolled out, cut strips of dough and then weave them into a lattice on top. Trim any excess dough from the edges and crimp the sides of the 2 pieces of dough together
Brush crust with some egg white wash and sprinkle evenly and lightly with 1-2 teaspoons of sugar. Place pie in refrigerator to chill while oven is preheating to 425° F.
Bake at 425° F on the middle rack for 20 minutes. Reduce heat to 350°F and bake for 35 minutes more or until done. You should see steady bubbling and steaming coming through the lattice or vents when it is done. Let cool before serving.