Preheat your oven to 375°F (190°C) and prepare a baking dish by lightly greasing it.
In a mixing bowl, combine the sliced peaches with cornstarch, granulated sugar, and the saffron threads dissolved in a small amount of warm water. Mix gently until the peaches are coated thoroughly.
Spread the peach mixture evenly into the prepared baking dish, ensuring the peaches are layered smoothly and juices begin to bubble around the edges.
In a separate bowl, stir together the oats, flour, and brown sugar to create the crisp topping. Cut in the cold butter using a pastry cutter or your fingers until the mixture resembles coarse crumbs.
Sprinkle the crumble topping evenly over the peaches, covering the surface completely for an even bake.
Bake the peach crisp in the preheated oven for 40 minutes, or until the topping is golden brown and the filling is bubbling around the edges.
Remove from the oven and allow to cool slightly before serving. The aroma of saffron and vanilla will fill the kitchen as it bakes, and the topping should be crisp and deliciously crumbly.