Preheat your oven to 375°F (190°C). In a large bowl, gently toss the sliced peaches with granulated sugar, lemon juice, and the chopped basil. Spread the peach mixture evenly into a greased baking dish.
In another bowl, combine oats, flour, brown sugar, and a pinch of salt. Mix well to evenly distribute the dry ingredients.
Add the cold, cubed butter to the dry ingredients. Use your fingertips or a pastry cutter to mix until the mixture resembles coarse crumbs with some pea-sized pieces.
Sprinkle the crumble topping evenly over the peach filling, covering all the fruit to get a full, textured topping.
Place the baking dish in the preheated oven and bake for about 40 minutes, until the topping is golden brown and the peaches are bubbling around the edges.
Remove the crumble from the oven and let it cool slightly. Serve warm, optionally garnished with fresh basil or a scoop of vanilla ice cream for added richness.