To make crust in food processor: Stir water and yolk in a 1-cup measure or small bowl until blended. Pulse flour and butter until coarse crumbs form. With motor running, add egg mixture. Process just until dough leaves sides of bowl. Remove blade, then dough. By hand: mix water and yolk until blended. Put flour in a medium bowl. Cut in butter with a pastry blender or 2 knives until mixture resembles coarse crumbs. Add yolk mixture and stir with a fork until crumbs clump together to form a dough.
Gather dough into a ball, divide in half, flatten each half into a disk and wrap in waxed paper. Refrigerate 30 minutes or until firm enough to roll out.
Filling: Mix 1 cup of sugar and flour in a large bowl. Add peaches, sprinkle with lemon juice and toss with sugar mixture until evenly coated.
Heat oven to 375 degrees Fahrenheit. Have a 9-inch pie plate ready.
On a lightly floured surface with a lightly floured rolling pin, roll out half the dough to a 12-inch circle. Fit circle into pie plate.
Spoon filling into pie shell and dot with 2 tablespoons butter.
Roll remaining dough into a 10-inch circle. Cut with a pastry wheel or sharp knife in twelve 3/4-inch-wide strips. Arrange 6 strips across filling, using longest in center. Lay remaining strips on top in opposite direction, weaving them under bottom strips if desired. Trim ends to inner edge of pie plate. Press ends to edge of of bottom crust. Roll overhang of bottom crust up over edges of strips to form an even rim. Flute or crimp edge. Brush lattice with egg-white mixture, then sprinkle with 1/4 cup sugar.
Bake 45 to 50 minutes until pastry is golden-brown and peaches are tender when pierced. Remove to a wire rack. Cool at least 1 hour before serving.
NOTE: To peel peaches easily, immerse them in boiling water 20 to 30 seconds, then immediately plunge them into ice water to stop cooking. Peel off skin, with your fingers or a small knife.