Whisk together 1/4 cup olive oil, 1/4 cup lemon juice, 1 tablespoon dried oregano, 1 tablespoon dried basil, chopped green onions, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a shallow dish to make the marinade.
Add the chicken breasts to the marinade, turning to coat completely. Cover and refrigerate for 30 minutes up to 4 hours.
Preheat your grill or grill pan to about 400°F (medium-high).
While the grill heats, whisk together 1/4 cup olive oil, 1/4 cup lemon juice, 1 teaspoon dried oregano, 1 teaspoon dried basil, 1/2 teaspoon salt, and 1/2 teaspoon pepper in a small bowl to make the dressing.
In a large bowl, combine the chopped Romaine, sliced cucumber, chopped red onion, chopped green bell pepper, chopped tomatoes, and pitted Kalamata olives.
Grill the marinated chicken 5 minutes per side, or until no longer pink and cooked through. Discard any remaining marinade.
Let the chicken rest about 5 minutes, then slice into 1/2-inch slices.
Drizzle the prepared dressing over the salad and toss to combine, then sprinkle the crumbled feta over the salad.
Top the dressed salad with the sliced chicken and arrange the avocado slices before serving.