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In large, shallow non-aluminum baking dish or plastic bag, pour 1/2 cup Italian dressing over steak; turn to coat. Cover, or close bag, and marinate in refrigerator 3 to 24 hours. Remove steak from marinade, discarding marinade.
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Grill or broil steak and onion, brushing frequently with 1/4 cup dressing, until steak is desired doneness and onion is tender. Brush cut side of bread with remaining 1/4 cup dressing and grill or broil until golden. Immediately arrange cheese on bottom half of bread, then top with remaining bread.
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Wrap in aluminum foil and let stand until cheese is melted. To serve, thinly slice steak. Arrange lettuce on serving platter, then top with steak, tomatoes and onion. Slice bread and arrange on salad.