Preheat oven to 350°F (175°C) and lightly grease a 9-inch baking dish.
Wash, peel, pit, and slice the peaches into even wedges.
In a large bowl, combine peaches, granulated sugar, brown sugar, cornstarch, vanilla, cinnamon, nutmeg, lemon juice, and salt. Toss until evenly coated.
Transfer the peach mixture to the prepared baking dish and spread evenly.
In another bowl, mix flour, oats, brown sugar, granulated sugar, cinnamon, and salt.
Cut the cold butter into the dry ingredients using a pastry cutter or fork until coarse crumbs form.
Sprinkle the crumble topping evenly over the peach filling without pressing it down.
Bake for 40–45 minutes, or until the topping is golden brown and the filling is bubbling around the edges.
Let the crumble cool for 10–15 minutes before serving.
Serve warm with vanilla ice cream or whipped cream.