According to Alexander Neckam's De naturis rerum (ca 1180), the cockatrice was the product of an egg laid by a cock (a male chicken) and incubated by a toad (from wiki)
If your cock doesn’t spend its entire life in far bottom corner of the coop but has enough space around, once in a while, it may lay an unusual egg.
route A)
Take this egg and put it into container with toad, frog, lizard or centipede. After some time it will hatch out a cockatrice. It may have some unique features depending on which animal it was sat upon by.
route B)
Leave this egg inside the coop as it is to hatch out a baby cockatrice. After hatching it immediately kills all the chickens and starts to eat one every X hours. Succeeding in this Y times will make it grow.
Adult cockatrice spawns outside the coop or container as a normal animal but never despawns until killed. It is as strong as a bear, has venomous bites and a unique attack which can instantly KO people (I’d rather see insta-kill moves that turn people into stone statues, but meh, casuals).
After defeating can be butchered for “the death-darting eye of Cockatrice” curio, which also KO people who a dare to peer into it.