Chap eats first of radishes before cheese
I believe the answer is:
gruyere
'cheese' is the definition.
(I know that Gruyere is a type of Swiss cheese)
'chap eats first of radishes before' is the wordplay.
'chap' becomes 'Guy'.
'eats' is an insertion indicator.
'first of radishes' becomes 'r' (1st letter of 'radishes').
'before' becomes 'ere' (poetic term for 'before').
'guy' enclosing 'r' is 'gruy'.
'gruy'+'ere'='GRUYERE'
(Other definitions for gruyere that I've seen before include "Tangy cheese" , "Whole-milk Swiss cheese" , "Firm pale cheese with holes" , "Whole-milk holey cheese" , "Hard yellow Swiss cheese" .)