The Lady of Bilsdale is mentioned in Tom Scott Burns 1993 book “Cleveland Hills” where it is described as the ‘enigmatic face of a lady carved on a rock wall who gazes eternally down into Bilsdale’.
I have been unable to find any other references to the carvings existence, so the phrase ‘enigmatic’ is certainly true. For those unable to access the remote sandstone quarry above Hasty Bank I have made a 3D scan of the carving.