After an encounter card with the surge keyword is revealed, the player resolving the card reveals an additional encounter card from the top of the encounter deck immediately after resolving the original card.
When do you Surge when you reveal Exhaustion?
Surge is executed last. When Exhaustion is revealed, you have to exhaust your identity, if able. Then you execute Surge by drawing a new card from the top of the encounter deck, and reveal it like any other encounter card.