Preheat a cast-iron skillet, griddle or grill pan over medium-high heat. In a large skillet, heat a drizzle of EVOO over medium-high heat. Add the butter and melt until foaming. Add the mushrooms and cook until browned, about 10 minutes; season with sea salt and pepper. Stir in the shallots and thyme for 2 minutes. Stir in the marsala (or sherry) and cook until almost evaporated, about 5 minutes. Stir in the beef stock and keep at a low simmer.
Meanwhile,preheat the broiler and drizzle the steak with EVOO. Cook on the preheated skillet, turning once, for 12 minutes for medium-rare. Season the meat with sea salt and pepper and let rest for a few minutes. Thinly slice the meat against the grain.
Toast the cut sides of the split rolls under a hot broiler until deep golden. Rub with the halved garlic, then drizzle with EVOO. Layer the meat and mushrooms on the roll bottoms and cover with the roll tops.