In a bowl or jar, whisk together sesame oils, honey, tamari, rice vinegar, ginger, garlic, orange zest, and sesame seeds. Mix until fully combined and slightly thickened. Set aside so the flavors can blend while you prepare the salad.
Heat avocado oil in a skillet over medium heat. Add sliced wonton wrappers in small batches. Fry until golden and crisp, then drain on paper towels.
In a large bowl, combine cabbage, lettuce, chicken, nuts, and orange slices. Drizzle half of the dressing and toss gently to coat everything evenly. Add avocado slices and most of the green onions.
Top with crispy wonton strips. Drizzle remaining dressing over the salad. Add chili oil if desired and serve immediately for best texture.