Preheat oven to 375°F (190°C) and position rack in the center. Lightly grease a 9-inch deep-dish pie pan with butter or cooking spray.
If using a store-bought crust, let it sit at room temperature for 10-15 minutes. Unroll and gently press it into the prepared pie pan, leaving about 1 inch of overhang. Fold the overhang under to create a thicker rim, then crimp the edges. Place the pie shell in the freezer for 10-15 minutes.
Peel the peaches using a vegetable peeler or by blanching them in boiling water for 30 seconds then transferring to ice water. Slice into 1/2-inch pieces. In a large bowl, combine the peach slices with granulated sugar, cornstarch, lemon juice, vanilla, cinnamon, nutmeg, and salt. Toss gently to coat and let rest for 5-10 minutes.
In a medium bowl, whisk together flour, oats, brown sugar, granulated sugar, cinnamon, and salt. Cut in the cold butter using a pastry cutter until the mixture resembles coarse crumbs with pea-sized pieces. Refrigerate while assembling the pie.
Pour the rested peach filling into the chilled pie shell, spreading evenly. Sprinkle the cold crumble topping evenly over the entire surface, pressing down very lightly.
Place the pie on a rimmed baking sheet lined with parchment paper. Bake at 375°F (190°C) for 45-55 minutes, until the topping is deep golden brown and the filling is bubbling around the edges. Cover loosely with foil if browning too quickly.
Remove from oven and let cool on a wire rack for at least 30-45 minutes before slicing. This allows the filling to set properly. Serve warm or at room temperature.