Preheat the oven to 350°F (175°C). Lightly grease a 9×9-inch baking dish.
In a large bowl, combine the sliced peaches, granulated sugar, brown sugar, lemon juice, vanilla extract, cornstarch, cinnamon, and nutmeg. Toss until the peaches are evenly coated.
Spread the peach mixture evenly into the prepared baking dish.
In another bowl, combine the rolled oats, flour, brown sugar, cinnamon, and salt. Pour in the melted butter and stir until the mixture becomes crumbly.
Evenly sprinkle the crisp topping over the peaches without pressing it down.
Bake for 40–45 minutes, or until the topping is golden brown and the peach filling is bubbling around the edges.
Let the peach crisp cool for about 10 minutes before serving. Enjoy warm with vanilla ice cream, whipped cream, or caramel sauce.