For the crust: Whisk the flour with the sugar and salt in a large bowl. Toss 4 pieces of butter with the mixture to coat, and then use your fingers to rub the butter into the flour until the mixture is powdery and pale yellow. You should not see any obvious bits of butter, and the mixture should feel cool and dry to the touch. Work in the remaining butter in the same manner, but this time just until pieces are the size of large beans. Combine the water and vinegar in a liquid measure and drizzle it evenly over the flour mixture. Toss and stir the mixture with a rubber spatula until it comes together in a loose, shaggy dough. A large handful of dough should hold together when you squeeze it in your palm; if it does not, add another tablespoon or so of cold water and toss again.
Turn the dough onto a clean surface and shape into a rectangle with a short end toward you. Starting at the far end, press the heel of your hand into the dough and smear it away from you with short strokes. Continue pressing and smearing, working to the edge close to you. When all the dough has been worked, scrape and press it together in a mass and divide into 2 equal pieces. Shape pieces into disks and wrap each in plastic wrap. Refrigerate at least 1 hour or up to 2 days.
Roll one piece of dough into a 14-inch circle and center it in a 10-inch pie plate; trim the overhang to 1 inch and refrigerate.
For the filling: Peel, halve, and pit the peaches. Slice each half into 6 wedges and halve them crosswise. Toss fruit with the sugar, tapioca, lemon juice, vanilla, and mace in a large bowl. Set aside until the peaches get juicy, about 15 minutes.
Roll the other piece of dough on a lightly floured surface to 1/8-inch thick. Cut into 1-inch-wide strips to create the lattice topping for the pie. Brush the rim of the bottom crust with some of the egg wash. Pour the fruit into the crust. Lay 5 to 7 strips 1/2-inch apart in one direction. Fold back every other strip halfway, and lay another strip in the perpendicular direction. Lay the strips back down. Fold back the alternate strips and lay a strip 1/2-inch from the first. Continue the process until the lattice is created. When the lattice is finished, trim the ends so that they meet the inside edges of the pie shell, tucking the lattice pieces under as necessary to create a finished look.
Brush the lattice with the egg wash, taking care not to drip over the sides, and trim the overhang flush with the edge of the pie plate. Fold overhang of bottom crust up and over lattice's edge; crimp and flute the edges as desired. Refrigerate pie for 30 minutes.
Position a rack as low as possible in the oven and preheat to 400 degrees F. Bake the pie on an aluminum foil-lined baking sheet until the crust is deep golden brown and the filling is bubbly, about 1 hour and 25 minutes. About halfway through baking, loosely crimp foil around the edges of the pie to protect the crust from burning and rotate the baking sheet. Cool the pie on a rack before cutting.