As the Sole Survivor arrives at the Museum of Witchcraft, the front doors are seen to be barred from the other side, requiring the use of a nearby cellar entrance in order to gain access.
Near the cellar, the corpse of a Gunner can be found, who holds a holotape that officially begins this quest. The holotape reveals that the Gunners sent a detachment to the area in order to fulfill a contract job, with a mysterious creature hunting them the entire way.
As the Sole Survivor progresses throughout the museum, a creature on the upper levels is heard feasting on the Gunners corpses, with body parts regularly falling through the decrepit floor boards.
Reaching the upper levels and unbarring the front doors, a savage deathclaw will suddenly attack the Sole Survivor, and must be put down.
After taking down the deathclaw, a holotape can be found on the corpse of a Gunners sergeant, revealing that their contract was to deliver a create full of broken deathclaw egg shells, as well as a pristine deathclaw egg, to a disclosed location. It is because of this egg, that the deathclaw was hunting their group down.
After picking up the egg, it can be returned to the mother's nest, where the corpse of another Gunner can be seen, alternatively it can be delivered to Welingham in Diamond city, for a reward of 200 caps (300/400/500 with speech checks). Afterward he will gift Wellingham's recipe believing the Sole Survivor to be a member of the Gunners who've agreed to deliver it somewhere.
Investigate the Museum of Witchcraft
Listen to Private Hart's holotape
I found a holotape on a mutilated body. Wonder what it says...
Explore the museum
A group of Gunners were chased by... something... into the Museum of Witchcraft. Maybe I can figure out what happened.
Complete delivery of the egg or Return the egg to its nest
These Gunners were delivering a case of Deathclaw eggs from their nest in Lynn Woods to Diamond City. I've got a decision to make - complete delivery or return the egg to its nest.
While the quest requires for the Sole Survivor to return the pristine deathclaw egg, the nest itself is guarded by other deathclaws, which must be dealt with in order to get close enough to the nest.
After the quest has concluded, the nest can be raided again for the pristine deathclaw egg. While the egg itself is unique, its value is the same, and counts as a regular egg at the food crafting station.