Complaint a vet is the man to deal with? (3,9)

'deal with?' is the definition.
I don't know anything about this answer so I cannot tell whether this works.

'complaint a vet is the man' is the wordplay.
I cannot quite understand how this works, but
'complaint' could be 'grievance' (grievance is a kind of complaint) and 'grievance' is located in the answer.
'the' could be 't' (the is pronounced as a 't' sound in some dialects) and 't' is present in the answer.
'man' could be 'p' (abbreviation for pawn, a chess piece or 'man') and 'p' is present in the answer.
A single letter 'e' remains which might be clued in a way I don't understand.
This may be the basis of clue (or it may be nonsense).

'to' is the link.

Can you help me to learn more?