Preheat a cast-iron skillet and a griddle or grill pan over medium-high heat. In the 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, garlic and thyme for 2 minutes. Add 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, for bread. Drizzle the steak with EVOO. Cook on the preheated cast iron griddle, 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 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.