Step 1: Preheat your oven to 375°F (190°C). Lightly butter a 9-inch square baking dish or a similar 2-quart oven-safe dish.
Step 2: In a large bowl, combine the sliced peaches, 1/2 cup granulated sugar, lemon juice, cornstarch, cinnamon, and nutmeg (if using). Toss gently until the peaches are well coated. Pour the peach mixture into the prepared baking dish, spreading it evenly.
Step 3: In a separate medium bowl, whisk together the flour, brown sugar, 1/4 cup granulated sugar, and salt.
Step 4: Add the cold, cubed butter to the flour mixture. Using your fingertips, a pastry blender, or a fork, work the butter into the dry ingredients until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining. Be careful not to overmix.
Step 5: Generously sprinkle the crumble topping over the peach filling, ensuring it's evenly distributed.
Step 6: Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbling around the edges. If the topping starts to brown too quickly, you can loosely tent the dish with aluminum foil.
Step 7: Remove from the oven and let cool on a wire rack for at least 15-20 minutes before serving. This allows the filling to set up slightly.