Add feta, yogurt (or sour cream), garlic, dill, and parsley to a food processor and blend on high until mixed. Wipe down sides as needed.
Transfer to a bowl to place on top of salad.
For the Grilled Chicken:
Combine the chicken, olive oil, garlic, thyme, oregano, and salt and pepper in either a shallow pan or large zip lock baggie. Coat chicken completely, cover, and refrigerate for 1-4 hours.
Preheat grill. Spray racks with cooking spray so items won't stick. Drizzle red onion slices with olive oil and season with salt and pepper.
Place chicken breasts on grill and cook for about 6 minutes. Flip and cook the other side. Place red onions on grill and flip once - they will have grill marks and become soft (you may remove early if needed and save for salad). Continue grilling until chicken reaches an internal temperature of 160° to 165°F. Remove from grill.
Combine chopped lettuce, tomatoes, and green onion in a bowl and drizzle with some olive oil and bit of lemon juice. You could choose any favorite salad additions. Toss to coat.
Serve salad on plate. Add feta mixture, grilled onions, and top with sliced chicken. Enjoy!