For steak marinade: In a screw-top jar, combine oil, vinegar, mustard, and oregano. Cover and shake well. Place steak in a reseable plastic bag and pour marinade over steak slices. Seal bag; turn to coat steak slices; set in a shallow dish and marinate in the refrigerator for at least 6 hours or up to 24 hours, turning bag occasionally.
Preheat broiler. Drain steak slices, discarding marinade. Slash fat edges of steak at 1-inch intervals, being careful not to cut into the meat. Place steak on the rack of an unheated broiler pan. Broil 3 inches from heat for 15 to 17 minutes or until medium doneness (160 degree), turning once halfway through broiling. At this point season with salt and pepper, if desired. Cool slightly; cut into thin slices.
To serve: Arrange shredded lettuce in a chilled large bowl. Place steak slices, tomatoes, avocado, cheese, bacon, and eggs on top of the lettuce. Drizzle some of the Dijon Vinaigrette over salad.
To make Dijon Vinaigrette: In a small bowl, white balsamic vinegar or white wine vinegar; Dijon mustard; garlic, minced, sugar, salt and pepper. Add olive oil in a slow steady stream, whisking constantly.