This is a guide for those who missed their chance on the story mission.
There is a way to find more braincage dragons. You need to use the reset garrisons option (pause-options-gameplay-reset garrisons) to get them, so make sure you've done everything you need to using garrisons (capture them all, use as teleport points for collectible finding, etc).
Once you've done this there are a couple of garrisons you can use, I suggest the one that is north-east of the dam at the far west of the map. (coordinates 476.7, 470.3). There is a dragon with a brain cage right outside this. Shoot this off and kill the dragon if you want. Then you can reset the garrisons and the dragon will be back, with it's braincage on.
Repeat 5 times for the achievement.
Resetting garrisons is also a good way to get money for upgrades, as you can go round and capture them all again.
EDIT:
Curtieson adds that you don't even need to reset garrisons to do this. Go to the garrison at (476.7, 470.3) and shoot the braincage off the dragon. Then fast travel to another garrison nearby, and drive back to (476.7, 470.3). The dragon should be back, with braincage on. Repeat 5 times for the achievement.