Make the filling: Stir together peaches, cornstarch, sugar, salt, cinnamon, nutmeg, and lemon juice. Set aside.
Preheat oven to 425°F (220°C) with a rack in the lower third.
Roll out pie crust into a 12" circle, transfer to a 9" pie dish, trim excess, and crimp edges. Spoon peach filling into crust, leaving some liquid behind. Refrigerate while making topping.
Make crumble topping: Whisk flour, brown sugar, granulated sugar, cinnamon, and salt. Stir in melted butter until crumbly. Sprinkle over peaches.
Bake at 425°F for 15 minutes, then reduce to 350°F for 45-50 minutes. Cover edges if browning too quickly. Let cool 3 hours before serving.
Notes
For the best flavor, use fresh, ripe peaches. Store leftovers in the refrigerator for up to 3 days.