In the future, when you get to 40, you are directed by a quest to the choco stables at Bentbranch Meadows. You'll need your retired chocobo form and a covering permission grade 1 of the opposite gender. The result is grade 2. Repeat with a grade 2 covering permission and your retired grade 2 bird, etc etc.

Keep in mind, you can only breed your retired bird 10 times. Also, while the covering proof is unique, you can hold more than one (use saddlebag and retainers). By doing so, you can breed your choco multiple times all at once (busy boy!). The ones that don't have good star ratings, you send off to the choco glue factory.