Preheat the oven to 350 degrees F.
With a sharp knife cut the peaches in half. Hold the peach vertical and cut around the peach. Take each side of the peach in the palm of your hands and twist.
Remove the pit. Place the peach halves in a baking dish.
Make the crumble topping. Add the flour, oats, brown sugar, and cinnamon to a medium bowl and mix together. Cut the butter in until everything is combined.
Divide the crumble into four equal parts. Place the crumble on each of the peaches. Gently press the crumble on top of the peach.
Place in the oven and bake for 25 to 35 minutes or until the peach is soft.
While the peaches are baking make the caramel sauce.
In a small saucepan add the butter, brown sugar, light corn syrup, and heavy cream. Cook over medium heat until comes to a boil. Reduce heat to medium-low and continue to cook for two to three minutes or until it is slightly thickened. Remove from the heat and stir in the vanilla.
When the peaches are done remove them from the oven and set them on a cooling rack. Serve them warm with the warm caramel sauce.