Shape the ground beef into 4 oblong patties, about 4 inches long each. Drizzle the patties with a little olive oil and sprinkle with salt and pepper; repeat on the opposite sides. Cut the french bread into pieces a little shorter than each of the patties and set aside.
Brown the patties in the skillet on both sides until the burgers are medium rare. Transfer the burgers to a plate and keep warm; leave the juices in the pan.
Melt 1 tablespoon of butter and 1/2 tablespoon of oil in the skillet. Add the shallot and garlic. Saute until fragrant, about 30 seconds.
Add another 1/2 tablespoon of the olive oil, if necessary, then add the mushrooms and saute until tender and the juices evaporate, about 3 minutes. Season with salt. Add the Marsala. Simmer until the Marsala reduces by half, about 2 minutes. Add the broth and the rosemary sprig. Simmer until reduced by half, about 4 minutes. Return the burgers to the skillet. Cook burgers until they are medium, turning to coat, about 1 minute. Season the sauce with salt and pepper, to taste. Add 1 oz of cheese to the top of each burger and cover the pan to allow the cheese to melt, about 2 minutes.
When the cheese has melted, remove the burgers from the pan and place on the French bread. Top the burgers with Marsala sauce and then the top of the French bread. Serve hot.