To make the pie dough

1. In a medium bowl, stir together 2 1/4 cups of the with the sugar and salt. Add the grated butter. Use a spatula to toss the butter to coat with the flour. Using a pastry cutter, cut the butter into the flour mixture, working quickly until pea sized pieces of butter remain -- do not overwork.
2. In a liquid measuring cup, stir together water, ice and vinegar. Sprinkle 2 tablespoons of the ice water mixture over the flour mixture. Continue to use the pastry cutter to cut the water into the flour mixture until it is fully incorporated. You want the dough to be sticky but not overly wet. Some of the pieces should still crumble off.
3. Use your hands to divide the dough in half. Shape one half into a round and the other half into a square. Cover with parchment paper and refrigerate for 1 hour. (The dough can be made ahead of time and refrigerated for up to three days or frozen for up to one month.)
4. Once chilled, remove the dough round from the fridge and cover the counter with flour. Roll the dough out into a round measuring 11 to 12 inches in diameter and 1/4 inch thick. Transfer the rolled-out dough to a 9-inch dark-coated pie plate and press it firmly into the bottom and sides. Use kitchen shears to trim the excess dough from the side of the pie plate, leaving 1/4 inch of overhanging dough over the rim. Cover loosely with plastic wrap and freeze while rolling out the remaining dough.
5. Re-flour the counter and roll the second square of dough into a large rectangle 1/4 inch thick. Use a pastry wheel or sharp paring knife to cut the rectangle into seven to eight 2-inch-wide strips. Transfer the dough strips to a parchment-lined baking sheet and freeze while preparing the filling.
6. To make the pie: In a large bowl, combine the peaches, blueberries, sugar, cornstarch, lemon zest and juice, vanilla paste, lavender extract, and cinnamon. Toss until the fruit is evenly coated with the sugar and seasonings. Let sit for 5 minutes.
7. Use a spoon to transfer the pie filling into the cold pie dough base, leaving any remaining juice behind in the bowl. Make sure the filling is as flat as possible.
8. To make the lattice weave, place a four pie strips in a row on top of the filling, about 1 inch apart. Weave the remaining strips perpandicular to the first rows, starting from the bottom and moving upwards, placing each strip about 1 inch apart. Use kitchen shears to trim off any overhanging dough, and then use a fork to crimp the lattice weave to the pie dough base, moving all the way around the edge of the pan. Cover loosely with plastic wrap and freeze for 1 hour before baking.
9. Place a rack just below the center of the oven and heat the oven to 400 degrees. Line a baking sheet with aluminum foil.
10. In a small bowl, whisk the egg with 1 teaspoon water until very smooth. Use a pastry brush to brush the egg wash over lattice and edges of the chilled pie. Sprinkle with additional sugar. Place on the prepared baking sheet.
11. Bake for 20 minutes. Check the crust and cover the top of the pie with aluminum foil if the edges are browning quickly. Lower the oven temperature to 350 degrees and continue to bake until the crust is golden brown and the filling is bubbling, about 15 minutes. Let the pie cool completely on a cooling rack before serving.