Make the Dressing: Combine basil, parsley, chives, garlic, anchovy (if using), lemon juice, and yogurt in a blender. Blend until smooth. With the blender running, slowly stream in the olive oil until emulsified. Season with salt and pepper. Set aside.
Bread the Chicken: Set up a breading station with three shallow dishes. In the first, mix flour with salt and pepper. In the second, place beaten eggs. In the third, combine panko, Parmesan, garlic powder, and oregano.
Dredge each chicken cutlet in flour, then egg, then firmly press into the panko-Parmesan mixture to coat. Place on a wire rack.
Cook the Chicken: Heat 1/4 inch of oil in a large skillet over medium heat. Once hot, add chicken (in batches if needed) and cook for 3-4 minutes per side until golden brown and cooked through (165°F internal temperature). Transfer to a clean wire rack.
Assemble: In a large bowl, toss the romaine, cucumber, tomatoes, and red onion with a few tablespoons of the Green Goddess dressing. Divide among plates. Top with sliced avocado and sliced Parmesan chicken. Drizzle with extra dressing and serve immediately.