Preheat the oven to 350°F (175°C). Grease a baking dish with butter or non-stick spray.
In a large mixing bowl, combine sliced peaches, granulated sugar, lemon juice, and cinnamon. Toss gently until the fruit is evenly coated, then pour the mixture into the prepared baking dish.
In a separate bowl, mix together flour, oats (if using), brown sugar, and salt. This forms the crumbly topping.
Add the cold, cubed butter to the dry ingredients. Using your fingertips or a pastry cutter, work the butter into the mixture until it resembles coarse crumbs with some pea-sized pieces.
Sprinkle the crumble topping evenly over the peaches in the baking dish.
Bake in the oven until the topping is golden brown and the peaches are bubbling around the edges, about 35 minutes.
Remove from the oven and let it cool slightly before serving. The crumble should have a crisp, golden topping with warm, soft peaches underneath.