After the dungeon maintenance and important documents quests, you will see some notes sparkling on the desk in the headquarters. Inspect them to enable a quest to reteieve an egg. Talk both NPCs to accept and advance it, then head outside to board the airship and start it. After a boring free fall featherfall session which you can skip with the I Am Stuck -Defeat Me option, you'll be tasked with finding the nest. Upon finding it the giant bird will knock you away. You will need to find 3 feathers then head down to the giant to have him fight the bird, allowing you access the nest and egg to complete the mission.
It lets you do the mission twice this first week for 6/15. You will need to repeat it for a few weeks. Alternatively, you can suicide after finding the 3 feathers and use the portal to exit, abandoning all progress. Then restart the mission, collect 3 feathers, and suicide/abandon progress again. Rinse and repeat a few times to unlock this right away if you are impatient.