Preheat the oven to 400°F (200°C). Fit the unbaked pie crust into a 9-inch pie dish, trim the overhang to ½ inch, fold under, and crimp the edges. Place the crust in the refrigerator while you prepare the filling.
In a large bowl, combine the sliced peaches, granulated sugar, flour, lemon juice, cinnamon, nutmeg, and salt. Toss gently to coat without breaking the peach wedges. Let the mixture sit for 5 minutes to draw out juices.
Pour the peach filling into the chilled pie shell, spreading it into an even layer. Leave behind any excess liquid pooled at the bottom of the bowl to ensure a non-runny filling.
In a medium bowl, whisk together the flour, brown sugar, oats (if using), cinnamon, and salt. Cut in the cold cubed butter using a pastry blender or your fingers until the mixture has pea-sized and slightly larger clumps.
Sprinkle the crumble topping evenly over the peach filling, covering it completely. Gently press down on any large clumps so the topping sits close to the fruit. Place the filled pie on a rimmed baking sheet.
Bake at 400°F (200°C) for 20 minutes. Reduce the oven temperature to 350°F (175°C) and continue baking for 35 more minutes, until the crumble is deep golden brown and the filling is bubbling at the edges. Cover the crust edges with a shield if browning too quickly.
Transfer the pie to a wire cooling rack and let it cool at room temperature for at least 3 hours before slicing to allow the filling to set properly.