Add both flours along with one teaspoon salt to the bowl of a food processor. Quickly pulse two times or until well blended. Add the cold cubed 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 the dough comes together and starts to form a shaggy ball. If necessary, add an additional two tablespoons of cold water, one tablespoon at a time, to bring the dough together. Be careful not to overprocess.

Turn dough out onto a work surface; knead until dough just comes together. Once again, do not overwork the dough. 

Divide the dough into two 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 for at least two hours or up to overnight.

Preheat the oven to 375°F.  

Combine peaches, brown sugar, cornstarch, vanilla bean paste, cinnamon & remaining 1/4 teaspoon salt in a large bowl; toss well to combine. Set aside.

Remove dough disks from the refrigerator and unwrap. Place larger dough disk between 2 lightly floured sheets of parchment paper or plastic wrap. Roll into a very thin 19- x 13-inch rectangle. Using a thin pan or cutting board for support, carefully slide dough from plastic wrap onto an 17- x 11-inch rimmed baking sheet. Remove plastic wrap, and press dough into corners of the baking sheet. 

Spoon peach mixture evenly over dough. Place second (smaller) dough disk between 2 lightly floured sheets of plastic wrap; roll into a very thin 11- x 17-inch rectangle. Cut dough lengthwise into 3/4-inch-thick strips. Arrange strips in a lattice pattern over peach mixture. Crimp edges to seal, using excess dough if necessary.  

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.