Place chicken in a large, zip top bag. In a medium bowl, whisk together 1 cup of salad dressing and lemon juice. Add to bag. Seal and marinate chicken at least 2 hours, up to overnight.
Heat grill to medium-high heat. Grill chicken until cooked through, about 5 minutes per side. For the last couple minutes of cooking, add the flatbreads to the grill and cook until marked and slightly crispy.
Let chicken rest for 5 minutes. Meanwhile, combine the romaine lettuce, additional Caesar dressing and Parmesan cheese and toss to mix well. Slice the chicken on an angle.
Add some of the salad mixture to each flatbread. Top with sliced chicken. Serve immediately.