When a Unique Asset is discarded is it shuffled back into the pack as with other double sided cards or are they just put into a discard pile?

Discard pile. Everything you use it is put into a discard pile(separate for each type of card) that gets reshuffled if need be.

Rules Reference Guide page 5: Double-sided cards, such as Spells or Conditions, are immediately shuffled back into their respective decks when discarded.

This Guide is based on the base game and Unique Assets didn't exist at that point but this rule also applies for Unique Assets, shuffle them back into their pile if discarded.

Technically you shuffle them back into the deck, as Roger mentioned above, however at this point we have so many of each type I tend to have a discard pile. If the card I need isn't in the active deck, I shuffle the discard pile and then draw from that.