Cook rotini in water with the tablespoon salt until fully tender. Drain, rinse briefly under cool water and let drain thoroughly while the chicken cooks.
Season cutlets with 1/4 teaspoon salt and pepper. Heat chicken oil in a skillet over medium heat and cook for 4 to 5 minutes per side until at least 165°F; rest on a clean board for 5 minutes.
Whisk dressing oil, lemon juice, vinegar, oregano, mustard and remaining salt in a large bowl. Add drained pasta, cucumber, tomatoes and olives and toss gently.
Slice the rested chicken and fold into the pasta with the feta. Let stand for 5 minutes so the dressing coats the ingredients.
Serve promptly, or refrigerate within two hours for a chilled meal; toss once more before portioning so the dressing is evenly distributed.