Landenel asks you to obtain the egg of a giant adamantoise for Gegeruju's banquet.
This quest is available if you have completed the quest A Modest Proposal.
Starting the Quest
Talk to Landenel in the South Shroud to begin the quest. There will be a cutscene.
Lay Hands Upon the Adamantoise Egg
Go east from Camp Tranquil to reach the orange area marked on the map. Check on the purple glowing Adamantoise Egg in the upper-right of the orange circle. Kill the Adamantoise that appears.
Claim the Adamantoise Egg
Check on the Adamantoise Egg to take it.
Present the Egg to Landenel
Go west and up the stairs to the area with the Camp Tranquil Aetheryte, and talk to Landenel to hand over the Adamantoise Egg.
Speak with U'odh Nunh at Forgotten Springs
Go to Southern Thanalan. If you are starting from the northern part of Southern Thanalan (e.g. near the Little Ala Mhigo Aetheryte), then use the small southwest bridge, then start going southwest to the end of the Red Labyrinth area. Talk to the Flame Private at the end, and you will be taken to the other door on the map. Go south through the tunnel to reach the Forgotten Springs. (If the Flame Private doesn't let you through, you will need to instead go south through the middle of Southern Thanalan, because the small tunnels on the left are just dead ends.)
In Forgotten Springs, attune to the Aetheryte, then go into the southwest building and talk to U'odh Nunh to complete the quest.
Next Main Scenario Quest
After you complete the quest, you will need to be at least level 31 to proceed to the next quest. After you reach level 31, talk to U'odh Nunh to accept the next quest: The Perfect Prey.