Preheat your grill or grill pan to medium-high heat.
Cut the cabbage into thick wedges, keeping the core intact so the leaves stay together.
Brush the cabbage wedges and the chicken breast with half of the olive oil, then season both sides evenly with sea salt, black pepper, and garlic powder.
Place the chicken on the grill and cook for 6-7 minutes per side, or until the internal temperature reaches 165°F.
Add the cabbage wedges to the grill and cook for 4-5 minutes per side until the edges are deeply charred and the center is slightly tender.
While the food grills, whisk together the remaining olive oil, lemon juice, chopped parsley, and chopped dill in a small bowl.
Slice the grilled chicken into strips and serve alongside the charred cabbage wedges, drizzling the fresh lemon-herb dressing over everything before serving.