In a small bowl, mix together the garlic, basil, and pepper. Rub over the turkey breasts. Insert one clove into each end of the turkey breasts, and one in the center.
In a large shallow dish, blend vegetable oil, soy sauce, lemon juice, and brown sugar. Place the breasts in the dish, and turn to coat. Cover, and marinate in the refrigerator at least 6 hours.
Preheat grill for high heat.
Lightly oil the grill grate. Discard marinade, place turkey breasts on the grill. Close the lid, and grill turkey breasts about 15 minutes on each side (unless it is a TEC, then reduce time by 40%), or to an internal temperature of 170 degrees F (68 degrees C).