Put the chicken, garlic, rosemary, and 2 tablespoons olive oil in a medium bowl. Season with salt and pepper and toss to combine. Cover and let marinate in the refrigerator for 4 hours or up to overnight. Remove the chicken from the refrigerator.
Put the nectarines in a small bowl. Add 1 tablespoon olive oil and gently toss to combine. Set aside.
Add the balsamic vinegar, honey, and 2 tablespoons olive oil to a small bowl. Season with salt and pepper and whisk to combine. Set aside.
Prepare a grill to medium-high heat. Brush the chicken on both sides with the honey and balsamic mixture. Put the chicken on the grill and grill for approximately 6 minutes per side, or until they are cooked through, brushing with the honey and balsamic mixture. Put the nectarines on the grill, cut side down, and grill for 4 minutes or until they are softened.
Remove from the grill. Transfer the chicken and nectarines to a platter or individual plates. Serve immediately.