Quoted about once becomes big-headed (9)
I believe the answer is:
conceited
'big-headed' is the definition.
Both the definition and answer are adjectives.
Perhaps you can see a link between them that I can't see?
'quoted about once' is the wordplay.
'quoted' becomes 'cited' ('cite' can be a synonym of 'quote').
'about' indicates putting letters inside.
'cited' placed around 'once' is 'CONCEITED'.
'becomes' acts as a link.
Can you help me to learn more?
(Other definitions for conceited that I've seen before include "Stuck-up" , "Vain; narcissistic" , "full of himself" , "Vain, having swelled head" , "Excessively vain" .)