Piaf's way to feel regret (3)
I believe the answer is:
rue
I believe this is a double definition.
'piaf's way' is the first definition.
('road' or 'way' in French, language of Edith Piaf)
'feel regret' is the second definition.
(I've seen this before)
'to' is the link.
(Other definitions for rue that I've seen before include "Bitter-tasting herb" , "The 'herb of grace'" , "Wish not to have been" , "Swann's way?" , "Bitterly regret - plant" .)