When advancing to the next stage of the quest, do we reshuffle the encounter deck? There is a rule on one of the stages, players must find and defeat a certain monster, but that monster is already in the discard pile from the previous stages. Would I just shuffle that card back in or shuffle the whole discard pile? Thanks!

The only times you shuffle the encounter deck is when told to do so by a card effect or when you run out of cards while dealing them during the quest phase.

Which quest are you questioning in particular?

*corrected*

Not during encounter, but during quest phase.

If by a chance OP is referring to Ungoliant Spawn from first adventure of core set, then you have to Find and kill it again when chosen quest card tells you so.

Of course you can just choose it as this one spider enemy that you have to put in Staging area...