Preheat oven to broil.
In a large sauté pan over medium-high heat, heat 2 tablespoons EVOO and 2 tablespoons butter. When the butter is melted, add the sliced onions, bay leaf, thyme, salt and freshly ground black pepper and cook until caramelized, about 15 minutes, or until melted down, brown and sweet.
While the onions are cooking, place steak on a broiler pan and generously coat it with EVOO, salt, and freshly ground black pepper. Broil under the broiler for 10 minutes or until cooked to desired doneness. When done, remove and let the steak rest to let the juices return to the center of the meat, about 5 minutes.
In a medium-size pot over high heat, boil water. Once at a boil, add the green beans and cook for 5 minutes, until cooked through.