1) Marinate the steak. Poke the steak all over with the tines of a fork; cut in half, against the grain, to make 2 steaks (one steak may be slightly thicker). In a dish large enough to hold the steaks, combine the olive oil, wine vinegar, shallot, garlic, and the thyme. Season with pepper to taste. Add the steaks and turn to coat evenly. Cover, and marinate the steaks at room temperature for 1 hour, turning once, or refrigerate, for up to 12 hours.
2) For the mushroom sauce: Heat a medium saute pan over high heat, add the butter, and when the foam subsides, add the mushrooms. Reduce the heat to medium-high and cook, the mushrooms, tossing frequently, until browned, about 5 minutes. Season with the 1/2 tsp. salt and pepper to taste. Add the garlic, shallot, thyme, and rosemary, and cook for 2 minutes more. Pour the red wine into pan, scraping up any browned bits on the bottom of the pan with a wooden spoon. Cook until almost all the liquid has evaporated. Add the cream, parsley, and chives and stir to combine. Set aside.
3) Heat a large grill pan over medium-high heat or an outdoor grill to medium-high. Remove the steaks from the marinade and pat dry. Season with salt and pepper. Cook the steaks for 3 to 5 minutes on each side for rare to medium-rare. Set aside on a cutting board for 5 minutes. Cut the steak against the grain on an angle into very thin slices. Arrange the slices on a warmed platter or plates. Reheat the mushroom sauce if needed and add any accumulated meat juices. Pour the sauce over the steak and serve immediately.
Cook's Note: Flank steak is the original cut for a London Broil.