Combine the vinegar and sugar in a small, nonaluminum saucepan. Bring to a boil over high heat on the grill or stovetop. Reduce heat to low and simmer, stirring ocassionally, for 30 to 45 minutes, or until glaze is thickened and reduced to about 1/3 cup. (Glaze may be prepared up to 3 weeks in advance; cover and refrigerate. Bring to room temperature prior to using.)
Lightly rub cut sides of peach halves with oil. Grill, cut side down, over a medium fire until slightly charred, 4 to 5 minutes. Brush cut sides with the glaze. Grill cut side up, for one minute.
Remove peaches from the grill to a cutting board. Brush on another layer of glaze, cut into 1/2-inch-thick slices, sprinkle with mint and serve warm.