For the record, here's the verse you get when you start one of the egg quests. In case anyone wants to give it a look (I copied it from the LL NPC, and I'm from Ul'dah, so, btw):

From a land not our own, a hero will come
To answer the song the Archons have sung.

And with his strong hands, an egg he will grasp
To grant it his warmth, for this is his task.

Yet one will not serve, as all lack a mother.
Twelve grant him strength to take up another.

A Twelve-fold vision to the Dreamer is due,
And from thence to the eggs, a spell she'll imbue.

When magicks have stirred the soul deep within,
The egg must be passed, and prayers begin.

Till flames of the heart see all rime effaced,
The Dreamer shall wait while the circle is paced.

The first three stanzas appear to just be talking about getting eggs, holding onto them, and getting more.

The fourth stanza could indicate getting the Pristine hat from the eggs, or it could indicate that there's something more the Dreamer can somehow do if we figure out how to trigger it. It could also just be talking about the imbuing of the elemental energies into the eggs before we're given them.

The fifth stanza could again indicate the crafting of the Pristine hat, the passing of the archon and elemental eggs to the NPC, or some other effect that we haven't seen yet.

The sixth stanza seems to talk about something we haven't figured out yet. "...the circle is paced" could talk about an actual physical route to take, an order of NPCs to talk to, or something else. I'm actually not sure.

At this point I should probably go back to the Lodestone and see if there's anything else I can glean....