Dropping a line to one of those on The Board - the one picking the players (7,8)
I believe the answer is:
casting director
'one of those on the board the one picking the players' is the definition.
I can't tell whether this definition defines the answer.
'dropping a line' is the wordplay.
'dropping' becomes 'casting' ('cast' can be a synonym of 'drop').
'a line' becomes 'director' (I can't justify this - if you can you should give a lot more credence to this answer).
'casting'+'director'='CASTING DIRECTOR'
'to' is the link.
Can you help me to learn more?
(Other definitions for casting director that I've seen before include "I assign players roles" , "Film executive" .)