Marinate blueberries in balsamic vinegar for 10-15 minutes. This can be done in advance so that you can pack ahead of time.
Layer sandwiches by spreading ½ T mayonnaise on one side of bread. Top with 2 slices of cheddar, 2 T marinated blueberries, 2 slices of mozzarella, and another piece of toast with ½ T mayonnaise on the inside.
Tightly wrap sandwiches with foil and take to park or camp grounds. Throw the foil packets on the grill or over an open fire until cheese is melted.