In a food processor or blender, blend yogurt, Parmesan, basil, parsley, lemon juice, vinegar, mustard, and honey until smooth; season with salt and pepper.
Make Ahead: Dressing can be made 3 days ahead. Store in an airtight container and refrigerate.
Generously season chicken with salt and pepper on both sides. In a large skillet over medium heat, heat oil. Add chicken and cook, turning occasionally, until an instant-read thermometer inserted into thickest part registers 165°, about 8 minutes per side. Transfer to a cutting board. Let cool 10 minutes, then slice.
In a large bowl, toss arugula, kale and lettuce mix, and romaine. Top with bacon, avocado, tomatoes, and onions and toss to combine.
Divide salad between bowls. Top with chicken and eggs. Drizzle with dressing.
