In a small bowl, whisk together the minced lemongrass, fish sauce, honey, minced garlic, and grated ginger.
Place the chicken breast in a shallow dish and coat with half of the lemongrass mixture, reserving the other half for the dressing.
Let the chicken marinate for at least 20 minutes in the refrigerator.
Preheat a grill or grill pan over medium-high heat and cook the chicken for 6 to 7 minutes per side until the internal temperature reaches 165°F.
While the chicken cooks, combine the shredded cabbage and carrots in a large mixing bowl.
Create the dressing by whisking the remaining marinade with the toasted sesame oil, rice vinegar, and lime juice.
Toss the cabbage and carrots with the dressing, fresh mint, and cilantro.
Slice the grilled chicken into strips and serve immediately over the cold, crunchy slaw, garnished with chopped peanuts.