Preheat the oven to 350°F (175°C).
In a large mixing bowl, combine the sliced apples, brown sugar, granulated sugar, lemon juice, cinnamon, nutmeg, cornstarch, and vanilla extract. Toss until the apples are evenly coated.
Transfer the apple mixture to a 10-inch cast iron skillet and spread it into an even layer.
In a separate bowl, combine the oats, flour, brown sugar, cinnamon, and salt.
Pour in the melted butter and mix until crumbly.
Sprinkle the topping evenly over the apples.
Bake for 40 to 45 minutes, or until the topping is golden brown and the apple filling is bubbling around the edges.
Remove from the oven and allow the crisp to cool for 10 minutes before serving.
Serve warm.