In a small sauté pan over medium heat, combine the vinegar and sugar. Add the grapes and cook to soften them, about 2 minutes, stirring occasionally. Transfer to a small bowl and cover to keep warm. Add the mint just before serving.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Unroll and rinse the grape leaves. Spread out the leaves on a work surface and pat dry with paper towels. Cut off and discard the tough stems. Overlap 4 grape leaves, vein-side up, into an 11- or 12-inch circle. Put another leaf in the center, and then put the cheese on top. Cover the cheese with another grape leaf. Wrap the leaves around the cheese, overlapping them to prevent the cheese from ultimately leaking out. Knot together two 3-foot pieces of butcher’s twine in the middle. Lay on a work surface in the shape of an X. Set the cheese bundle on top. Bring the twine to the top and tie snugly. Wrap the twine around the cheese 1 or 2 more times like spokes on a wheel, knotting snugly in the center each time. Tuck in any loose leaf edges. Trim the ends of the twine.
Lightly brush the cheese bundle and bread slices all over with oil. Brush the cooking grates clean. Grill the cheese over direct medium heat, with the lid closed as much as possible, until each side is soft when gently pressed, 3 to 4 minutes, turning once. Carefully remove the cheese and let rest for about 2 minutes. After you remove the cheese, grill the bread slices over direct medium heat until they are lightly toasted on one side only, about 1 minute.
Arrange the cheese and grilled bread on a serving plate or tray. Cut the twine and discard it. Pull the grape leaves open to reveal the cheese inside. Be sure to do this just before serving as the cheese begins to ooze out as soon as you remove the grape leaves. Top the cheese with some of the grape salsa or serve it alongside in a bowl. Use spoons to scoop up cheese and salsa, and place on slices of grilled bread.
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