Cook the Pasta: Boil pasta in salted water according to package instructions until al dente. Drain and rinse under cold water to stop cooking.
Prepare the Chicken: If not already cooked, season chicken breasts with salt and pepper, and pan-fry in a skillet until golden and cooked through. Let cool, then cube or shred.
Make the Dressing: In a bowl, whisk together mayonnaise, sour cream, sun-dried tomato oil, Dijon, garlic, basil, red pepper flakes, salt, pepper, and Parmesan.
Combine Everything: In a large mixing bowl, toss together cooked pasta, chicken, sun-dried tomatoes, cherry tomatoes, red onion, greens (if using), and dressing until well coated.
Chill: Refrigerate for at least 30 minutes to let the flavors meld.
Garnish & Serve: Top with more grated Parmesan or fresh herbs before serving, if desired.