Eastern stagehand covering books in exercise in self-aggrandisement (3,4)
I believe the answer is:
ego trip
'exercise in self-aggrandisement' is the definition.
(exercise or action which pleases one's ego)
'eastern stagehand covering books' is the wordplay.
'eastern' becomes 'e'.
'stagehand' becomes 'grip' (term for a theatre stagehand).
'covering' means one lot of letters goes inside another.
'books' becomes 'OT' (Old Testament).
'grip' placed around 'ot' is 'gotrip'.
'e'+'gotrip'='EGO-TRIP'
'in' acts as a link.
(Other definitions for ego trip that I've seen before include "Act done to increase one's self-importance" , "Experience enjoyed by the conceited" , "self-centred indulgence" , "Activity boosting one's self-importance" , "Self-indulgent activity" .)