Heat closed contact grill or panini maker 5 minutes.
Meanwhile, on cutting board, carefully unroll dough. Cut into 8 rectangles. Place 2 rectangles on grill; close grill. Cook 1 to 2 minutes or until lightly browned. Remove from grill. Repeat with remaining rectangles.
Spread 1 tablespoon barbecue sauce on each of 4 rectangles. Layer turkey, onion and cheese over sauce. Top with remaining rectangles.
Place 2 sandwiches on grill; close grill. Cook 2 to 3 minutes or until golden brown and cheese is melted. Repeat with remaining sandwiches. If desired, serve with mayonnaise or mustard.