I just finished the covert ops mission to unlock a Chosen stronghold mission, however my best soldiers are recovering from their wounds so I would rather not do the mission right now if possible. The text/voice stuff in the game seemed to imply that the mission is urgent and I need to it ASAP. Is this the case?
Yes and No
No they are not like other urgent missions in that the opportunity to undertake the mission will disappear if not taken immediately.
Yes they are urgent since the longer you take to undergo the mission the more information the chosen agent will gather on you, the commander. The more the bar is filled the more skills they unlock until they are finally able to assault you, the commander, directly.
That being said: It is perfectly reasonable to allow your A team to recover before sending them on this mission. As long as the Chosen agent you are about to take out isn't about to complete their information gathering.