Unwell following Greek restaurant (5)
I believe the answer is:
grill
'restaurant' is the definition.
(grill is a kind of restaurant)
'unwell following greek' is the wordplay.
'unwell' becomes 'ill' ('ill' can be similar in meaning to 'unwell').
'following' is a charade indicator (letters next to each other).
'greek' becomes 'gr'.
'ill' after 'gr' is 'GRILL'.
(Other definitions for grill that I've seen before include "Cooking device" , "Question hard" , "Intensely question; dish" , "Seriously question" , "Grating" .)