Should I Expect a Discount to Replace a Broken Veneer Done Only a Year Ago?

Unless it was something that you specifcially did to break the veneer, it is reasonable to expect not to pay full fee to replace a broken veneer done a year ago.  I am not familiar witht the specific details of your treatment, so this is a generalization.  There may be more to your story.  In my office unless the patient did something to break the veneer, I just replace it.  If they broke it themselves even by accident, I still usually reduce the fee for work done that recently.  There are a lot of other factors including quality of the lab used, initial fees, experience of the dentist, etc....

In my office I do offer a warranty for five years on veneers as long as they maintain their oral hygiene in my office and they do not cause the damage themselves such as biting on a pencil or something like that.

Every dentist is free to warranty their work as they see fit.  Porcelain can fracture on occasion.  In our office we usually will replace failed restorations for up to 5 years at no charge if patient has followed regular checkups and any protective measures such as wearing a nightguard if a night time grinding habit is present.