Start your grill on smoke with the lid open until a fire is established in the burn pot (3-7 minutes). Preheat to 450F.
Lay four 12 x 12 inch pieces of foil out on a flat surface, then place one chicken breast in the middle of each foil.
Divide the broccoli and string beans between the four foil packs. Thinly slice the lemon, split them between each foil pack, and place the slices on, in and around the chicken and vegetables.
Mix the butter, garlic, juice of the remaining lemon, and Italian seasoning together, and then brush over the chicken and vegetables. Sprinkle with salt and pepper to taste.
Fold the foil over the chicken and vegetables to close the pack, and pinch the ends together so the pack will remain closed.
Grill for 7-9 minutes on each side. Turn off grill, remove the foil packets, and serve immediately.