In a large skillet over medium heat, cook steak in 1 tablespoon olive oil for 4-6 minutes on each side or until meat reaches desired doneness (for medium-rare, a meat thermometer should read 145°; medium, 160°; well-done, 170°). Remove from the skillet; let stand for 5 minutes before slicing.
Meanwhile, in the same skillet, saute mushrooms until tender. In a small bowl, combine vinegar and thyme. Whisk in walnut oil and remaining olive oil. Stir in walnuts.
Divide salad greens and shallot between two serving bowls. Cut steak into slices. Top salads with steak and mushrooms. Drizzle with dressing; sprinkle with cheese. Yield: 2 servings.