(n.) the point at which a terrible danger first becomes apparent
While a cockatrice is a hideous mythical hybrid, figuratively the expression cockatrice’s egg is used to refer to the point at which a terrible threat or danger first becomes apparent.
As you’ll doubtless already know if you know your myths and legends (or, for that matter, your Harry Potter), a cockatrice is a fabulous cross between a rooster and a serpent. According to legend, cockatrices were traditionally said to be born from an egg laid by a cock, not a hen, and incubated by a toad or a snake. The only way to stop the egg from hatching into the monster inside it was to throw it over a roof, and have it land on the other side without it once touching the house. Should the egg hatch, the cockatrice would be able to strike anyone it stares at stone dead with its glance alone; the only way to kill it would be to force it to see—or else trick it into looking at—its own reflection.
Its name (which is thought to be a confusion of the French coq, ‘cockerel’, and the Latin calcare, ‘to tread’) has been traced back to the late fourteenth century in English, but it can be found even earlier in its native French Stories of the fabled cockatrice, meanwhile, stretch back into antiquity.
As for the cockatrice’s egg, as an allusive reference to the starting point of a fatal threat or danger, it dates from the early 1600s.