Combine ground beef, 2 Tablespoons shallots, 2/3 of the minced rosemary, Worcestershire sauce, salt, and pepper in a large bowl then mix until combined. Create 4 burger patties then set aside.
Heat extra virgin olive oil in a large, cast iron or heavy bottomed skillet. Add burger patties then saute for 4-5 minutes a side for burgers cooked to medium. Add swiss cheese slices then allow to melt slightly. Transfer to a plate then loosely tent with a piece of nonstick-sprayed foil.
Melt 1 Tablespoon butter in skillet then add mushrooms and remaining shallots. Season with salt and pepper then saute until mushrooms are golden brown, 5 minutes. Add remaining rosemary and garlic then saute for 1 more minute. Add beef broth then simmer, scraping the bits up from the bottom of the pan, until sauce has reduced and thickened slightly, 5 minutes. Add remaining Tablespoon butter then stir until melted. Place burgers on buns then top with mushrooms and sauce.