This gooey mess of caramelized onion and barbecue sauce also tastes great as a cold sandwich. (And did you know caramelized onions freezes well? Make extra for topping burgers, pizzas and more.)
In a large skillet, cook onion, covered, in hot butter over medium-low heat for 13 to 15 minutes or until onions are tender, stirring occasionally. Uncover; cook and stir over medium-high heat for 3 to 5 minutes more or until golden. Remove from heat; cool slightly.
Meanwhile, in a small bowl, combine mayonnaise and barbecue sauce. Spread mixture evenly on the bread slices. Assemble sandwiches, layering cheese, ham, tomato and the caramelized onion.
Cook sandwiches on a preheated, lightly greased electric sandwich press, covered indoor grill, grill pan or skillet. If using a sandwich press or covered indoor grill, cook about 6 minutes or until toasted. If using a grill pan or skillet, weigh sandwiches down and cook about 2 minutes on each side or until toasted.