In a small glass bowl, whisk together the extra virgin olive oil, lime juice, grated ginger, minced garlic, sea salt, and black pepper to create the base marinade.
Place the chicken breast in a shallow dish and pour half of the prepared marinade over it, turning to coat thoroughly, then marinate for 15-20 minutes.
Preheat a grill or cast-iron grill pan over medium-high heat until hot.
While the chicken marinates, combine the shredded green cabbage, red cabbage, shredded carrots, and chopped cilantro in a large mixing bowl.
Whisk the remaining half of the marinade with the toasted sesame oil and rice vinegar to create the salad dressing.
Grill the chicken for approximately 6 to 7 minutes per side, or until the internal temperature reaches 165°F and beautiful char marks appear.
Transfer the chicken to a cutting board and allow it to rest for 5 minutes before slicing into thin strips across the grain.
Toss the cabbage mixture with the sesame-lime dressing until well coated and serve the warm chicken strips immediately on top.