"It seems after you have finished your company quest(~level 20), you can instant level 15 by talking to these NPCs as the correct class.
Swaenhylt in Camp Drybone(Eastern Thanalan) for Thaumaturge
Wihtred in Hyrstmill(North Shroud) for Conjurer
Aniud in Hawthorne Hut(East Shroud) above the town for Lancer
Chechezan in Horizon Camp(Western Thanalan) for Gladiator
Zacheus in Little Solace (East Shroud) for Archer
Urswyrst in Moraby Drydocks(Lower Noscea) outside gate near Chocobo Porter for Marauder
Adelard at the South Hammer(Western Thanalan)(20,27) for Pugilist"
There's a full list here: http://www.reddit.com/r/ffxiv/comments/1gubii/instant_level_15_on_jobs/