Heat a large skillet over MED/MED-HIGH heat. Add butter and olive oil and sliced onions. Season with 1/4 tsp pepper. Add thyme sprigs and cook 8 minutes, stirring very occasionally.
While onions are cooking, preheat oven to 400 F degrees and butter a 9x13" baking dish. Set aside.
After 8 minutes, reduce the heat on the pan of onions down to MED/MED-LOW, and cook another 15 minutes or so, stirring occasionally.
Stir in garlic, cook 1 minute. Stir in balsamic vinegar and 1/3 cup of the beef broth. Stir and cook another minute or so, until onion mixture is thick and jam-like in consistency.
Add about a 1/2 cup of the caramelized onions to the prepared baking dish and add the remaining 1/3 cup beef broth to the dish. Stir.
Season chicken breasts on both sides with salt, black pepper, dried thyme and garlic powder. Arrange chicken in prepared baking dish and top each breast with a generous spoonful of caramelized onions.
Top chicken and onions with mozzarella and parmesan.
Bake 25 minutes, or until chicken is cooked to an internal temperature of 165 F degrees.