Brett Kavanaugh is one of the coauthors of "Law of judicial precedent" that was published in 2016. Aside Law of judicial precedent, I do not think Brett has co-authored any other book or written anyone all by himself. He is known for his excellent writing skills and styles.
He wrote articles about basketball and other sports for Yale Daily News when he was at Yale University. He served as a Notes Editor for the Yale Law Journal. He also writes articles in national leading law journals. Brett has written many articles over the years but I do not think he has written a book.