Chase cool writer (8)
I believe the answer is:
inscribe
'chase' is the definition.
(chase can mean to etch or engrave)
'cool writer' is the wordplay.
'cool' becomes 'in' ('in' can mean fashionable or cool).
'writer' becomes 'scribe' (scribe is a kind of writer).
'in'+'scribe'='INSCRIBE'
(Other definitions for inscribe that I've seen before include "Engrave; dedicate" , "Write or carve on a surface" , "Engrave or write upon" , "Draw (a figure) within another" , "Engrave; enter (name in list)" .)