Preheat your oven to 375°F (190°C). Lightly grease an 8x8 inch baking dish or a 9-inch round pie plate.

Prepare the filling: In a large bowl, gently toss the sliced peaches with the granulated sugar, cornstarch, lemon juice, and 1/2 teaspoon of cinnamon until evenly coated. Pour the mixture into the prepared baking dish.

Prepare the topping: In a separate medium bowl, whisk together the flour, rolled oats, brown sugar, 1/2 teaspoon of cinnamon, and salt.

Cut in the butter: Add the cold butter cubes to the dry ingredients. Use your fingers, a pastry blender, or a fork to cut the butter into the flour mixture until coarse, pea-sized crumbs form. Do not overmix; some chunks of butter are desirable for crispiness.

Assemble and bake: Sprinkle the oat crumble topping evenly over the peach filling. Place the dish on a baking sheet (to catch any potential spills) and bake for 35 to 40 minutes, or until the topping is golden brown and the peach filling is vigorously bubbling at the edges.

Let the crumble cool for at least 15 minutes before serving. This allows the filling to set slightly. Serve warm, optionally with vanilla ice cream or whipped cream.