In a large bowl whisk flour, salt, and 1 tablespoon sugar. Cut in cold butter with a pastry cutter or two forks until mixture resembles coarse crumbs with some pea-sized pieces. Sprinkle 4 tablespoons ice water over the flour mixture and gently toss with a fork; add more water, 1 tablespoon at a time, until dough just holds when pressed. Divide into two discs, wrap in plastic, and chill at least 30 minutes.

Bring a pot of water to a boil, score the bottom of peaches, and blanch 20–30 seconds. Transfer peaches to an ice bath; slip off skins, then slice into 1/3- to 1/2-inch wedges and place in a large bowl. Add 1/3 cup granulated sugar, 1 tablespoon brown sugar, 1 tablespoon lemon juice, and 1 teaspoon vanilla; toss gently and let macerate 20–30 minutes. Drain peaches in a sieve for 10 minutes and reserve juices.

Whisk together 2 tablespoons cornstarch, 1 tablespoon quick-cook tapioca, 1/2 teaspoon cinnamon, 1/8 teaspoon nutmeg, and a pinch of salt. In a small saucepan combine reserved peach juices (add water to total 1/2 cup if needed) and the thickener mixture. Cook over medium heat, whisking, until mixture boils and thickens about 1–2 minutes. Remove from heat and stir in 2 tablespoons butter until melted. Toss thickened syrup with drained peach slices and set aside to cool slightly.

Preheat oven to 375°F. Roll one chilled dough disc into a 12-inch circle and fit it into a 9-inch pie dish; trim excess leaving a 1/2-inch overhang. Chill crust in the refrigerator 15 minutes. Line crust with parchment and fill with pie weights or dried beans. Bake on a baking sheet for 12 minutes, remove weights and parchment, and bake 5 more minutes until set but not fully browned.

Increase oven to 400°F. Spoon peach filling into the partially baked crust, mounding slightly. Roll top crust to 10–11 inches, place over filling, trim and crimp edges, and cut vents or create a lattice. Brush with egg wash and sprinkle coarse sugar if using. Place pie on a baking sheet and bake at 400°F for 20 minutes, then reduce heat to 375°F and bake another 25–30 minutes until golden and bubbling. Tent edges with foil if they brown too quickly. Cool at least 2 hours before slicing.

Serve warm or at room temperature with vanilla ice cream or whipped cream. Store covered at room temperature up to 2 days or refrigerate up to 4 days. Freeze fully cooled pie, wrapped tightly, up to 2 months; thaw overnight in the refrigerator and re-crisp in a 350°F oven for 10–12 minutes.