First to admit gluttony having consented (6)
I believe the answer is:
agreed
'consented' is the definition.
(agreeing is a kind of consenting)
'first to admit gluttony' is the wordplay.
'first to' says to take the initial letters.
'gluttony' becomes 'greed' (I've seen this before).
The initial letter of 'admit' is 'a'.
'a'+'greed'='AGREED'
'having' is the link.
(Other definitions for agreed that I've seen before include "Geared (anag.)" , "matched" , "very well" , "came to the same conclusion" , "Acceded" .)