Pulse parsley and garlic in a blender or food processor just until finely chopped. (Do not puree.) Remove to a medium bowl, and stir in lemon juice, olive oil, 3/4 teaspoon salt, and 3/4 teaspoon pepper; set aside.
Cook onion and 1/2 teaspoon salt in hot vegetable oil over medium-high heat, stirring often, 15 minutes or until onions are golden brown and tender.
Sprinkle steaks evenly with remaining 1/2 teaspoon salt and remaining 1/2 teaspoon pepper.
Grill, covered with grill lid, over medium-high heat (350° to 400°) 7 to 10 minutes on each side or to desired degree of doneness.
Cut rolls in half lengthwise, and grill cut sides of rolls during the last few minutes of cooking steaks. Remove steaks and rolls.
Cut steaks into thin slices. Spread parsley mixture evenly on cut sides of bread; place steak slices and onion evenly on bottom bread halves, and top each evenly with cheese and remaining bread halves. Wrap each sandwich in foil. Grill, covered with grill lid, 3 to 4 minutes or until cheese melts.