Preheat your oven to 375°F (190°C). Lightly grease a 9x9 inch baking dish or a similar oven-proof casserole dish.

Prepare the filling: In a large bowl, gently toss the sliced peaches with the granulated sugar, cornstarch, lemon juice, and cinnamon until the peaches are evenly coated. Pour the mixture into the prepared baking dish.

Prepare the crumble topping: In a separate medium bowl, whisk together the flour, rolled oats, brown sugar, and nutmeg.

Cut in the butter: Using a pastry blender or your fingertips, cut the cold butter cubes into the dry ingredients until the mixture resembles coarse crumbs, ensuring some pea-sized pieces of butter remain.

Assemble and bake: Sprinkle the crumble topping evenly over the peach filling in the baking dish. Bake for 35 to 40 minutes, or until the topping is golden brown and the fruit filling is bubbling around the edges.

Let the crumble cool slightly on a wire rack for at least 15 minutes before serving. Serve warm, optionally topped with vanilla ice cream or fresh whipped cream.