Place flour, sugar, and salt into the bowl of a food processor. Process for for 1 minute. Add the cold, diced butter, and pulse the machine several times until clumps form (do not let the dough mass on the blade). Butter must stay ice cold so place in freezer for a while if it gets soft.

Dump the crumbly mass onto your work surface, and roughly form it into a ball. Cut off 1/3 for lattice top, form each part into a disk, and wrap each disk in plastic. Chill in the fridge for 30 minutes.

Combine the sugars, cinnamon, nutmeg, tapioca and salt into a medium bowl, and whisk to combine. Put the prepared peaches and lemon juice in a large bowl, pour on the sugar mixture, and toss to coat. Let the juices exude for at least 10 minutes.

Unwrap one disk of dough, and roll it into a 12-inch-diameter circle. Place the circle in a standard 9-inch pie plate and freeze. Then roll the second crust into a circle, cut into strips and freeze. When strips are solid, form lattice. Add the filling and all of the accumulated juices to chilled pie plate and cover with lattice .Brush the top with the beaten egg, and sprinkle with sugar.

Center the oven rack, and preheat the oven to 425°F. Set the pie on a parchment-lined baking sheet, and bake for 20 minutes. Then lower the heat to 375°F and bake until the crust turns golden-brown, and the juices bubble -- 35-40 minutes. Cool on a wire rack for 3 hours. Then -- to insure a firm, not runny pie -- chill in the fridge for several hours or overnight.