In a sauté pan over medium-high heat, combine the pitted cherries, sugar, red wine, and balsamic vinegar. Bring to a simmer and cook, stirring occasionally, until the fruit is soft, 6 to 8 minutes. Transfer the mixture to a food processor and purée until completely smooth. Return the mixture to the sauté pan over medium-high heat. Add the kirsch. Simmer until reduced to about ¼ cup, 1 to 2 minutes.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Cut the peaches in half and remove and discard the pits. Place the halves in a medium bowl. In a small saucepan set over low heat, melt the butter and brown sugar together. Coat the peaches with the butter mixture.
Brush the cooking grates clean. Grill the peaches over direct medium heat, with the lid closed as much as possible, until grill marks are clearly visible and the peaches are soft, 8 to 10 minutes, turning once.
While the peaches are still warm, layer each serving glass with two peach halves, one scoop of ice cream, and 1 tablespoon cherry sauce. Tuck a cookie into each glass. Serve immediately.