Season chicken with its salt and pepper. Heat oil in a skillet and cook cutlets for 4 to 5 minutes per side until at least 165°F, then rest for 5 minutes and cut into bite-size pieces.
Meanwhile, boil penne in water with the tablespoon salt until fully tender. Drain, rinse under cool water and drain thoroughly. Spread chicken in a shallow dish to stop steaming.
Whisk mayonnaise, Parmesan, lemon juice, Dijon, anchovy paste, garlic and water until smooth. Toss pasta and cooled chicken with the dressing.
Fold in dry romaine and croutons just before serving. To make ahead, refrigerate the dressed pasta and chicken promptly and keep lettuce and croutons separate until the final toss.
Serve cool and refrigerate leftovers within 2 hours.