Combine first 3 ingredients; rub evenly over steak. Cover and chill 2 hours.
Place steak on a broiler pan coated with cooking spray; broil 7 minutes on each side or until done. Cut steak diagonally across grain into thin slices.
While steak cooks, heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add onion to pan; cover and cook 10 minutes or until golden brown, stirring frequently.
Combine cheese and mayonnaise; spread 2 tablespoons on each lavash half. Top each with 3 ounces steak and 1/2 cup onion; roll up.