Preheat the oven to 350°F (175°C).
In a large bowl, combine sliced peaches, granulated sugar, cornstarch, vanilla extract, and ground cinnamon. Toss gently to coat the peaches evenly.
Transfer the peach mixture to a 9x9-inch (or similar size) baking dish, spreading it evenly.
In a separate bowl, mix oats, all-purpose flour, brown sugar, ground cinnamon, nutmeg, and salt.
Add the cold, cubed butter to the oat mixture. Using a pastry cutter or your fingers, work the butter into the dry ingredients until the mixture resembles coarse crumbs.
Sprinkle the crispy topping over the peach mixture in the baking dish. Place the baking dish in the preheated oven and bake for 35-40 minutes, or until the topping is golden brown and the peach filling is bubbly.
Once the peach crisp is baked, remove it from the oven and let it cool for a few minutes. Drizzle the warm peach crisp with the homemade caramel sauce. Serve each portion of peach crisp with a scoop of vanilla ice cream on top.