The act of feeding specific limit broken cards 
297 + Points and receiving an additional random awakening from a pool of one to eight awakenings depending on the card in question.
Super Awakenings are rolled at random, and thus, it may take you many attempts to get a desired awakening on a given card, especially when there are five or more in the pool of possible awakenings.
How to unlock Super Awakening
Quick guide by Reni:
Important Note
Super Awakenings are active only in solo gameplay, and are disabled in multiplayer gameplay unless all players are disconnected (which would then revert the dungeon back to solo gameplay).
More in-depth guide
For a more detailed look at Super Awakenings, see here!