Warm sesame oil in a heavy large skillet over high heat. Sprinkle steak with salt and pepper. Add steak to skillet and sauté until no longer pink, about 2 minutes. Add mushrooms, scallions and carrot. Stir-fry until vegetables are crisp-tender but still bright in color and steak is cooked through, about 2 minutes longer. Add teriyaki sauce and toss to blend well.
Arrange greens on dinner plates. Top with stir-fried mixture and sprouts. Serve immediately.