If a 1 or 2 year gap in the chronology of a resume is the result of
dealing with a pretty significant medical condition, how do you handle
that in the interview process?
You should say nothing about the time gaps unless asked or it is otherwise necessary to account for that time, or unless you need to ask for special accommodations due to your condition.
Even then, only explain as much as is necessary for the interview. Often you just need to acknowledge that there is a real gap, that it isn't a typo, and that you had something you needed to deal with. You might need to convey that you weren't just sitting around without a job for no reason.
Most employers won't care at all about the details of your medical condition, but some might need to be reassured that it is something which is in the past (if that's truly the case). Again, no need to recount any painful details - just enough to answer the questions being asked.
Go all the way and proactively explain what the condition is and that
the condition is almost certainly resolved, but still in an absolute
worse case scenario X might still happen?
This is something only you can judge.
If there is still a reasonable possibility of a recurrence or flare-up or something else that would mean significant time off or even leaving the job, some would say that it is best to be proactive about the condition.
You need to consider the likelihood, the impact, and the severity of the condition.
How would an official document from an elite medical specialist that
says you are cured come into play?
I would never resort to this unless it were a condition for employment (for example, if you couldn't be a pilot without medical clearance from your heart surgeon).