Mix garlic, basil, and pepper together in a small bowl; rub over turkey breasts. Insert 1 clove into each end of turkey breasts and 1 in the center.

Mix oil, soy sauce, lemon juice, and brown sugar in a large, shallow dish. Place turkey in the dish and turn to coat. Cover and marinate in the refrigerator at least 4 hours.

Preheat an outdoor grill for high heat and lightly oil the grate.

Discard marinade and place turkey on the preheated grill. Close the lid and cook until turkey is no longer pink in the center and the juices run clear, about 15 minutes on each side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).