Add butter and olive oil to a large skillet, heat for a minute over medium/high heat until the butter melts. Add the onions. Stir and cook the onions until they become translucent and slightly browned on the edges, about seven minutes. Add the balsamic vinegar, salt, and pepper. Sautee onions until deeply caramelized, about another ten to twelve minutes. Once caramelized, add chopped basil and turn off heat.

Warm a large skillet or griddle for two minutes over medium heat.

Butter one side of each slice of bread. Place the bread butter side down on the griddle, top each slice with about a half a cup of shredded cheese. Place half the caramelized onions on two slices of bread. After about three to four minutes flip the slice with only the cheese on to the top of the slice with the onions. Cook another minute or two until golden and melty.

Let sandwich sit for a minute or two before cutting or serving so the cheese has time to slightly harden. Serve immediately.