Place the peaches on the grill cut side down and cook for 5 minutes. Flip and cook for another 5 minutes. Remove from the grill and set aside.
Make the caramel sauce
In a heavy bottom sauce pan over medium-high heat add the sugar and water. Stir until the sugar has dissolved.
Bring to a boil and stop stirring. Allow the solution to cook until it starts turning a caramel brown. At this point do not stir but gently swirl the pan until the syrup is evenly a rich caramel color. This can take 4-8 minutes. Remove from the heat.
Add the cream, this will cause a violent reaction. Stir until the sauce is nice and smooth. Add the butter, vanilla, and salt and stir to combine. Add the bourbon or rum if you desire.
Serve the peaches with ice cream topped with the caramel sauce.