Employment of an American egghead (3)
I believe the answer is:
use
'employment' is the definition.
(how something is used)
'an american egghead' is the wordplay.
'an american' becomes 'US' (United States).
'egghead' becomes 'e' (head letter of 'egg').
'us'+'e'='USE'
'of' is the link.
(Other definitions for use that I've seen before include "Apply, employ" , "Exploit unfairly" , "Put to work" , "Exploit; custom" , "Resort to - operation" .)