Preheat oven to 375°F (190°C). Grease a 9x13-inch baking dish.
Make the crust dough: In a large bowl, mix flour and salt. Cut in cold butter until crumbly. Add ice water one tablespoon at a time until dough comes together. Divide into three portions, wrap, and chill 20 minutes.
Roll out first portion and fit into bottom of baking dish. Trim edges, prick with fork, and pre-bake 10 minutes.
Prepare filling: Toss peaches with granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, lemon juice, and vanilla.
Roll out second crust and place over pre-baked bottom layer.
Spread half the peach filling over second crust. Roll out third crust and place on top of remaining peaches. Crimp edges, cut vents, and brush with milk.
Bake 45-55 minutes until crust is golden brown and filling bubbles. Rotate pan halfway. Cool 15-20 minutes before serving.