How to Make It
Pulse flours and 1 teaspoon salt in a food processor 2 times. Add butter; pulse until mixture resembles coarse meal, 4 to 5 times. Stir together oil, lemon juice, and 2/3 cup cold water in a small bowl. While pulsing, pour oil mixture through food chute. Process just until dough comes together and starts to form a shaggy ball. Add an additional 2 tablespoons cold water, 1 tablespoon at a time, if necessary to bring dough together. (Be careful not to overprocess.)
Turn dough out onto a work surface; knead until dough just comes together, about 2 times. Divide dough into 2 pieces, one roughly 60% of the dough and the other 40%. Flatten each into a 1-inch-thick disk; wrap each in plastic wrap. Chill 2 hours or up to overnight.
Preheat oven to 375°F. Combine peaches, brown sugar, cornstarch, vanilla bean paste, cinnamon, and remaining 1/4 teaspoon salt in a large bowl; toss well to combine. Set aside.
Remove dough disks from refrigerator, and unwrap. Place larger dough disk between 2 lightly floured sheets of plastic wrap; roll into a very thin 20- x 15-inch rectangle. Using a thin pan or cutting board for support, carefully slide dough from plastic wrap onto a 13- x 18-inch rimmed baking sheet. Remove plastic wrap, and press dough into corners of baking sheet. Spoon peach mixture evenly over dough. Place second dough disk between 2 lightly floured sheets of plastic wrap; roll into a very thin 13- x 18-inch rectangle. Cut dough lengthwise into 3/4-inch-thick strips. Arrange strips in a lattice pattern over peach mixture; crimp edges to seal. Whisk together egg yolk and 1 teaspoon room-temperature water in a small bowl. Brush dough lattice evenly with egg yolk mixture.
Bake at 375°F until filling is thick and bubbly and crust is browned, about 50 to 55 minutes. Cool at least 20 minutes before serving.