Heat oil and butter in a saute pan over medium/high heat. Add onions once butter is melted. After 5 minutes, reduce heat to medium and give onions a stir. Continue to cook over medium heat, if onions seem to be leaning towards burning, turn down to medium/low. Stir occasionally but not too often. After about 1 minutes, season with salt, pepper and worcestershire sauce. Cook another 5-10 minutes until onions are soft and caramelized. Deglaze the pan with the beef broth and continue to cook until liquid is just about absorbed; approximately 5 more minutes. Set aside.
While onions are caramelizing, in a large bowl combine beef, worcestershire sauce, beef broth and black pepper. Shape into patties. You should get about 6 BIG patties. Be sure to press your thumb in the center of each patty, creating a divot, which will result in a more evenly rounded patty after cooking.
Heat grill to medium/high heat. Place burgers on preheated grill. Cook about 5-6 minutes until burger easily releases from grates. Flip over, reduce heat to medium to medium low, depending on how hot your grill cooks. Close cover and cook another 8-12 minutes or until desired doneness. Place cheese on each burger, close cover just until cheese melts, about 2 minutes.
Place buns, cut side down on grill to toast - about 3 minutes.
Create burger by placing cheeseburger on bun an top with caramelized onions.