In a medium skillet over medium–high heat on the stove, combine the cherries, wine, granulated sugar and balsamic vinegar. Bring to a boil and reduce the heat to medium–low. Simmer, stirring occasionally, until the fruit is tender, 6 to 8 minutes. Remove from the heat and let cool until tepid. Transfer the mixture to a food processor and purée until completely smooth. Return the mixture to the skillet. Stir in the kirsch and bring to a boil over medium–high heat. Reduce the heat to medium and boil, stirring often, until slightly reduced, about 2 minutes. Remove from the heat and allow to cool completely.
Prepare the grill for direct cooking over medium heat (180°C to 230°C).
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 on the stove, 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, until grill marks are clearly visible and the peaches are tender, 8 to 10 minutes, turning once. Remove from the grill and allow to cool slightly.
Brush the cooking grates clean. Grill the peaches over medium heat, with the lid closed, until grill marks are clearly visible and the peaches are tender, 8 to 10 minutes, turning once. Remove from the grill and allow to cool slightly.
To individual dessert bowls add two warm peach halves, a scoop of ice cream, some of the sauce and a rolled cookie. Garnish each with a mint sprig. Serve immediately.
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