Preheat oven to 450 degrees F. For large onions, peel and cut into wedges, leaving as much root on as possible. If using boiling onions, peel and leave whole. Heat olive oil in a large ovenproof saute pan until hot. Add onions, and reduce heat to medium. Cook until brown on all sides, about 5 minutes. Season with salt and pepper. Add garlic and cook briefly until light brown. Add thyme and stir. Regulate heat by moving pan around. Add 2 Tbs. of the vinegar and toss well with onions.

Place onions in oven and roast until tender and very browned, about 30 minutes. Stir gently at least once during roasting. Remove to a plate. Add remaining 2 Tbs. balsamic vinegar to pan and stir and scrape up bits. Pour over onions.

Heat the remaining 1 Tbs. of olive oil in a heavy saute pan over medium-high heat until very hot. Season the steak with salt and pepper. Turn on the exhaust fan and place steak in pan. Cook until brown on first side, about 2 minutes. Turn and place in oven to roast until done, about 4 minutes for rare meat. Just before removing, dot the top with butter.

Remove steak to a carving board and let rest for 5 minutes. Return onions to the saute pan and reheat by tossing over medium heat. To serve, cut the meat off the bone, then carve into thin slices. Arrange the meat on plates and top with onions. Pour any juices from the carving board over the meat.