Cook the pasta. Bring a large pot of salted water to a boil.
Cook pasta until just al dente, then drain and rinse briefly under cool water to stop the cooking. Shake off excess water and transfer to a large bowl.
Toast the pine nuts. In a dry skillet over medium heat, toast pine nuts for 2–3 minutes, stirring, until golden and fragrant. Remove to cool.
Make the dressing. In a jar or bowl, whisk the sun-dried tomato oil (or olive oil), lemon zest and juice, Dijon, garlic, oregano, red pepper flakes, a pinch of salt, and several grinds of black pepper until emulsified.
Toss the base. Add sun-dried tomatoes, cherry tomatoes, red onion, and olives to the pasta.
Pour over half the dressing and toss to coat.
Add the chicken. Fold in the cooked chicken and most of the basil and parsley. Add more dressing as needed to coat everything lightly without pooling at the bottom.
Finish and season. Gently mix in the toasted pine nuts and feta (or Parmesan). Taste and adjust salt, pepper, and lemon to brighten.
Rest, then serve. Let the salad sit for 10–15 minutes so the flavors meld.
Garnish with remaining herbs. Serve at room temperature or chilled.