In a grill, prepare a medium-hot fire for direct-heat cooking.  In a large bowl, combine chicken and a little salt and pepper.  Handling the meat as little as possible to avoid compacting it, mix well.  Divide the mixture into 6 equal portions and form the portions into round patties to fit the buns. 

When the fire is ready, brush the grill rack with vegetable oil.  Place the patties on the grill rack, brush the tops with barbecue sauce, and cook until browned on the bottom, about 4 minutes.  With a wide spatula, turn the patties and cook, brushing frequently with the sauce, until the juices run clean when the patties are pierced, about 4 minutes longer.  

During the last few minutes of cooking, place the buns, cut side down, on the outer edges of the grill to toast lightly and top each patty with a cheese slice to melt.   Transfer the patties to the bottom halves of the buns and cover with bun tops. 

Serves 6.