I still haven't understood what you need to do to get it.
Is it a quest that needs to be done? Or a collection of local quests? Or all quests in zone?
Its a few quests and a few ops: (wiki)
Grauberg - Brugaire's Ambition, The Egg Enthusiast and Grauberg Dominon Ops 1, 2, 3, 6
Kenshyn is right, those are the quests you need to do. The Egg Enthusiast requires level 1 of fame and Brugaire's Ambition requires level 3 of fame in Abyssea-Grauberg.
I personally did Koruo-Moruo's item delivery quest because it was the cheapest and fastest way to get fame in the area.
I bought one stack of each of the three possible items he may ask and used MM+Fleet Wind to speed things up, but I ended up using Spring Shoes because that saved me from calling Garuda, which was growing tiresome.
From what I noticed, all of the level 1 fame quests in Abyssea-Grauberg force you to zone upon completition in order to retry and most of them (excluding Egg Enthusiast and Koruo-Moruo's quest) require items that cost between 20-40k gil on my server per atempt.
After you have done at least once every quest and Dom. Op. listed by Kenshyn, you should be able to synth it for 1k with any Atma Fabricant in a Heroes area.
Also, the Brugaire' Ambition is an excelent quest but it requires a pop item that can be found in the pool of possible loots of the pots that you need to kill for one of the four Dom. Ops. I was lucky and got a drop and saved 20k myself that way.
Furthermore, I don't think a SMN can solo Brugaire's Ambition or at least, not easily, because the ??? position is found in a reduced area. Also, if you bring someone who can proc blue !! on it, you might get some Caller Seals:Hands out of it. A kind party that was killing it for seals offered me their SMN seals because nobody wanted them and between their popset and ours, I ended up with 4 (or 5?) Caller's seals:Hands! It might be worth of notice that the item Brugaire gives and asks you to use on the NM paralyzes it for a short amount of time with a potent paralyze, so it can be used as safety meassure too. Although a BLU+BLM+NIN can trio it easily without a problem, a WHM may be needed to Cure V or Cure VI from time to time if one of it wasn't possible to stun of its potent spells. Edited, Jan 5th 2011 12:45pm by AmrikValefor