I remember reading this English short story during my school days; it was part of our English curriculum 18-19 years ago.

Plot:

It starts with a wife killing her husband with turkey or chicken or some other food item : not a planned murder but one in anger during a fight. Then the murder investigation starts and a police constable tries to figure out who killed him, but he fails and didn't even get his hands on the murder weapon. In the end, the wife offers him a dish made out of the murder weapon and he starts eating it while either he or the wife says something like:

Sometimes evidence is just under your nose and you fail to find it.

While evidence was literally under his nose.

Genre: Murder Mystery/Drama

Lamb To The Slaughter.

This was a short story by Roald Dahl that was initially rejected.

She owes it to the child to escape discovery if she can. She prepares the leg of lamb that she used as a weapon and places it in the oven to somewhat destroy the evidence. Then she considers an alibi. After practising a cheerful mask and some innocuous remarks to make in conversation, she visits the grocer and chats blandly with him about what to make for Patrick's dinner. Upon her return to the house and to the room where her husband lies dead on the floor, she acts surprised and meaningfully cries. She then calls the police.
When the police (who are all friends of her husband) arrive, they ask Mary questions and look at the scene. Considering Mary above suspicion, the police conclude that Patrick was killed by an intruder with a large blunt object likely made of metal. After they make a fruitless search around the house and surrounding area, Mary is reminded that the leg is just about done and offers it to the policemen. She points out that they have already been working through and past the dinner hour, and that the meat will otherwise go to waste; they hesitate, but accept. During the meal, as Mary sits nearby but does not join them, the policemen discuss the murder weapon's possible location. One officer, his mouth full of meat, says it is "probably right under our very noses". Mary, overhearing them, begins to giggle.
_{^{- Lamb To The Slaughter - Wikipedia}}

This seems to match your description pretty well.