In a large, heavy skillet over medium-high heat on the stove, melt the butter. Separate the onion slices and add them to the skillet. Cook the onions until they begin to brown, about 10 minutes, stirring often. Reduce the heat to medium and cook until the onions are very tender and caramelized, about 15 minutes more, stirring often. Season with the salt and ¼ teaspoon pepper. Remove from the heat and allow to cool.
Prepare the grill for cooking over medium heat (400°F).
In a small bowl combine the mayonnaise, basil, and ⅛ teaspoon pepper.
Lightly brush one side of each bread slice with oil. Turn all the bread slices over and spread with the mayonnaise mixture, dividing equally. Top eight of the bread slices with equal amounts of turkey, onions, and cheese. Cover with the remaining eight bread slices, mayonnaise mixture side down. Lightly brush the top of each sandwich with oil.
Brush the cooking grates clean. Grill the sandwiches over medium heat, with the lid closed, until the cheese melts and the bread is golden, 3 to 5 minutes, turning once. Remove from the grill and serve right away.