Combine cheeses. Lightly butter sheet of aluminum foil 3-4 inches larger than tortillas. Place tortilla on middle of foil; sprinkle one-thired of the cheese over tortilla, then one-third of the scallions. Top with second tortilla and press lightly together. Fold foil loosely over edge of tortillas, leaving most of the top exposed. Repeat with remaining tortillas, cheese, and scallions.
When grill is waqrming and before it is hot enough to grill meat, place 1 or 2 foil packages, foil side down, on grill. (They can also be grilled under the broiler. Broil, foil side down, 6 inches from heat until cheese melts, about 4 minutes.) Grill until cheese melts, about 4 minutes. dlip packages and grill until top is lightly toasted, 1-2 minutes.
Immediately cut each quesadilla into 6 wedges and serve hot. Grill remaining quesadilas.