Season the chicken. Sprinkle the chicken breast with salt and black pepper on both sides. You don't need a heavy seasoning because the dressing will add plenty of flavor later.
Cook the chicken. Heat the oil in a skillet over medium heat. Add the chicken and cook for about 5–7 minutes per side, depending on its thickness. The thickest part should reach 165°F (74°C).
Let the chicken rest. Transfer the cooked chicken to a plate and let it rest for a few minutes. This helps keep the meat juicy when you slice it.
Slice the chicken. Once it has cooled slightly, cut the chicken into thin slices or small pieces. You can also shred it if you prefer a more traditional salad texture.
Prepare the vegetables. Add the shredded cabbage, shredded carrot, and sliced green onions to a large bowl. Toss them together so the vegetables are evenly mixed.
Make the dressing. In a small bowl, whisk together the soy sauce, rice vinegar, honey, sesame oil, ginger, and garlic. Keep stirring until the honey is mixed into the dressing.
Add the chicken. Place the sliced or shredded chicken into the bowl with the vegetables.
Dress the salad. Pour the dressing over the chicken and vegetables. Start with most of the dressing and add the rest according to your taste.
Toss everything together. Gently toss the salad until the chicken and vegetables are coated. Try not to mix too aggressively because you want the cabbage and other vegetables to stay crisp.
Add the crunchy toppings. Sprinkle the sliced almonds and crispy wonton strips over the salad just before serving. Adding them at the end keeps them from becoming soft.
Taste the salad. Give it a quick taste and add a little more dressing if needed. If you prefer more sweetness, a small drizzle of honey can also work.
Serve right away. The salad is best when the vegetables are crisp and the toppings still have their crunch.