The Order of the Blessing of Throats on the Feast of Saint Blaise is in the Book of Blessings (de Benedictionibus).
Article 1625 from The Book of Blessings contains a brief historical background of the blessing of throats:
Details regarding the miraculous healing of the boy vary. One account relates that the miracle occurred during the journey to take Blaise to prison when he placed his hand on the boy's head and prayed; another that the miracle happened while Blaise was in prison when he picked up two candles provided to him and formed a cross around the boy's throat.
The use of candles for the blessing of throats stems from the candles that Blaise used while in prison. When an old woman's pig had been miraculously rescued from a wolf by Saint Blaise, she would visit him in prison, bringing him food and candles to bring him light in his dark cell.
Articles 1626 and 1627 explains when and how the blessing takes place: