In a small bowl, combine 2 teaspoons Worcestershire sauce, 1 teaspoon hot pepper sauce, garlic and thyme. Rub steak with marinade. Cover and refrigerate at least one hour or overnight.
Preheat broiler. Spray rack in broiler pan with nonstick cooking spray.
Place steak on prepared broiler rack. Broil 3–5 minutes on each side or until desired doneness. Meanwhile, in a small bowl, combine the remaining 1 teaspoon Worcestershire sauce, 1⁄2 teaspoon hot pepper sauce and broth.
Remove steak from broiler, cover with aluminum foil and let stand for 10 minutes. Slice steak thinly across the grain; add juices that accumulate while slicing to Worcestershire sauce mixture. Pour mixture over steak and serve immediately.