Doctors that are specialized in taking on the Overflow Infection although they are not able to cure it since there is no cure. However, they have quite the knowledge about the infection that stems from old documents and research in prior ages, especially the Age of Construction during which overflow doctors became a thing. The Overflow Infection itself has been around for far longer, probably even since the Age of Draft but nobody is absolutely sure about that, so this is just a theory.
People in earlier days thought that the infection can be contagious which is why overflow doctors often wore helmets that were made out of the same material that people use for Everlasting Devotions and that had a distinguished look to them. The infection is not contagious at all, however, overflow doctors still kept their old style and even the ones of today usually wear said helmets, either out of tradition or because they like to stay anonymous since they are so high in demand and want some free time for themselves.
As said before, overflow doctors are not able to cure the infection but present-day medicine enables them to extend the lives of infected people by a few years, in some cases even over 10. Infected people that do not see an overflow doctor and do not get regular treatment usually only live up to one year, so it makes a big difference. Overflow doctors try to be fair when it comes to their prices even though they could demand much more since their operations are usually not covered by any healthcare and patients have to pay for the treatment themselves. Regularly visiting them can extend the rest of one's live but because of how they doctor people - they use measures to reduce Essence Energy inside people and other methods - most patients will notice that their Aptitude powers grow weaker, especially for Supplicants. This side effect is one of the reasons, next to having to pay for yourself, that not everyone who is infected visits them, even though that is most careless and will lead to a short life.
Overflow doctors also offer to take a patient's life should they be getting close to the final stage, so that others do not have to do it and to give them a quick, painless death. This “service” is free when you allow them to keep your body for experiments after your death. Overflow doctors hope to somehow find a cure this way but they are still far away from it. Nobody knows if it is even possible to find a cure.