Make the dressing: Add the orange zest and ¼ cup fresh orange juice, toasted sesame oil, coconut aminos, avocado oil, rice vinegar, ginger, garlic, and maple syrup to a jar with a lid. Shake well until combined. You can also whisk everything together in a small bowl.
Marinate the chicken. To a bowl, pour about half of the dressing over the chicken breasts and marinate for 20-30 minutes at room temperature while you prep the vegetables for the salad. Reserve the remaining dressing for tossing with the finished salad.
Assemble the salad. To a large bowl, add the Napa cabbage, red cabbage, shredded carrots, green onions, cilantro, drained mandarin oranges, and almonds. Add sesame seeds, if using.
Cook the chicken. Heat 1–2 tablespoons avocado oil in a large skillet over medium-high heat. Once hot, add the marinated chicken breasts and cook for about 5–7 minutes per side, depending on thickness, until cooked through and the internal temperature reaches 165°F.
Transfer the chicken to a cutting board and let rest for about 10 minutes. Roughly chop or slice, then add to the salad.
Pour the remaining sesame-orange dressing over the salad and toss until everything is well coated. Top with extra sliced almonds or sesame seeds for a little more crunch, if desired.