The Beak Thing Egg is an obscure Animal Product item. Eggs can only be found in the nests of Beak Things. They are decently valuable and can be sold to any trader. Due to the eggs' weight and space, it is difficult to carry many of them by one character.
A particularly good area to acquire these items is from Gut. Beak Thing Nests are easy to spot and plentiful in this environment, plus one is in close proximity to numerous cities that these can be sold to, such as Brink and Heng.
It is not possible to grow pets from the eggs, but during development the Devs have hinted that it may be implemented in future.