1. Heat a grill pan over medium-high heat. Rub thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper over steaks. Add steaks to pan; cook 4 minutes on each side or until desired degree of doneness. Remove steaks from pan. Add lemon halves, cut sides down, to pan; cook 3 minutes. Cut steaks across the grain into thin slices.
2. Combine remaining 1/4 teaspoon pepper, chives, and next 4 ingredients (through 1/8 teaspoon salt), stirring with a whisk. Drizzle over arugula; toss to coat. Arrange 1 steak, 1 cup arugula, and 1 lemon half on each of 4 plates; top each salad with 1 tablespoon cheese.