Over the years I've heard of a handful of Navy flight surgeons who also had their AME ticket and would issue an FAA medical, a Class 2 seemed to be the most common though. Usually you hear about them through word of mouth but you can also look up (online) FAA AMEs by location.
FAA medicals work a bit differently than military flight physicals. The mil flight pays is a little more cut and dried- you're in one or the other service group and it has a hard expiration date. FAA medicals start out at whatever class you checked off on the form on the day you got it done. After a certain period of time then they're not good for that class anymore but they still count for a lower class medical. They technically don't "revert" to a lower class (again, it's spelled out in governmentese). How long they last until they effectively revert depends on your age. The medical class required depends on the kind of flying you're doing and it's indirectly related to flying for hire and carrying paying passengers.