You choose which permanent cards to return while Eerie Ultimatum is resolving. No player may take actions between the time you choose and the time those cards return to the battlefield.

You may choose to return just one permanent card, regardless of its name.

If an Aura is put onto the battlefield without being cast, the Aura’s controller-to-be chooses what it will enchant as it enters the battlefield. An Aura put onto the battlefield this way doesn’t target anything (so it could be attached to an opponent’s permanent with hexproof, for example), but the Aura’s enchant ability restricts what it can be attached to. If the Aura can’t legally be attached to anything, it remains in its current zone.

An Aura being put onto the battlefield this way can’t enchant anything else that is being put onto the battlefield at the same time.