This is a "lighter" (i.e., lower in calories and fat) version of the classic New Orleans sandwich made with sliced deli meats.
Combine the olives, capers, oregano, and vinegar in a bowl and set aside.
Sprinkle the chicken breasts with the Cajun spice blend. Heat the oil in a large nonstick skillet set over medium high heat; add the chicken and cook until golden on both sides and cooked through, about 2 minutes per side.
Split the rolls and pull out the middle breading to hollow them out. Divide the olive mixture (solids only) among the four roll bottoms. Top with the romaine lettuce and the chicken. Brush the cut side of tops of the rolls with the liquid from the olive mixture. Place the cheese on the tops, and close the sandwiches.
Wrap each sandwich tightly in plastic wrap and place in the refrigerator, weighting them down lightly with a small bottle or container. Let sit from 20 minutes to overnight.