Prepare grill or broiler.
To prepare sauce, combine first 5 ingredients, stirring with a whisk.
To prepare steak, combine juice and next 4 ingredients (juice through steak) in a large zip-top plastic bag; seal. Marinate in refrigerator 10 minutes, turning once. Remove steak from bag; discard marinade.
Place steak on grill rack or broiler pan coated with cooking spray; cook 6 minutes on each side or until desired degree of doneness. Let stand 5 minutes. Cut steak diagonally across grain into thin slices. Line each pita half with 1 lettuce leaf half. Divide steak evenly among pita halves. Spoon 1 tablespoon sauce and 1 1/2 teaspoons cheese into each pita half.