Fire up. Set up your grill for direct heat cooking on medium high heat. When it is hot make sure to clean the grates thoroughly so there is no meat grease. This is crucial! If necessary, use water to loosen the grease.
Cook. While the grill is preheating, pour the rum, lemon juice, vanilla, maple syrup, brown sugar, butter, and salt into a sauce pan and melt them together over high heat, stirring occasionally. Boil the mix for about 10 minutes to drive off some of the alcohol. I know you think this is a waste, but you want the booze to play nice with everybody, not bully them. It really detracts from the taste in this recipe. When it is done, let it cool.
Prep. Cut the peaches in half and remove the pits.
Cook again. Grill the peaches cut side down until they get some nice golden grill marks but try not to let the sugars burn. Should take about 20 minutes. After about 20 minutes if you stab the skin side with a knife or a fork it should slide in easily. If not, flip them over and cook a bit longer on the skin side until there are grill marks. Remove them from the grill and cut them into slices or chunks.
Serve. Put a scoop of ice cream in a bowl, pour the sauce over it (a ladle comes in handy), and then top with the peaches. Serve immediately because the ice cream will start melting fast.