Go outside your starting nation, talk to the moogle. Kill bats with level cap in effect. Each bat you kill will, after talking to the moogle a second time, give you an en-light effect. 1 Damage per bat, caps at 30 IIRC. Use the Enlight to kill the big eye. Once dead, return to the moogle for your reward: Eerie Cloak.
To HQ, do the same thing only wear the cloak while killing.