Preheat your grill to 450 degrees F. Ensure your grill grates are clean and free of any other food remnants.

In a medium saucepan, combine the balsamic vinegar and cherry halves. Turn the heat to medium high and bring the mixture to a boil. Reduce the heat to low and simmer for 5-10 minutes, or until the vinegar has reduced to a slightly thin syrup consistency. Turn off the heat and set aside; the sauce will thicken as it cools.

Brush the cut sides of each peach with the melted butter. Place the cut sides of the peaches down onto the preheated grill grates and grill for 3-4 minutes (rotate the peaches after 2 minutes if you want some gorgeous grill marks on them). Carefully flip the peaches onto the skin side and cook for an additional 3-4 minutes. The peaches should have nice grill marks and be noticeably softer.

Transfer the peach halves to your serving platter and drizzle with the cherry balsamic sauce. Roll the basil leaves and slice them into a thin chiffonade and sprinkle on top of the peaches. Serve as is or with a scoop of vanilla ice cream for a delicious dessert.