The USAF has a dual-designator Flight Surgeon-Pilot option and the Navy has had one as well on and off though the years. I knew at least two USAF types, one was already a flight surgeon-pilot and the other had just gotten selected for flight school after being a flight doc for a few years, he had over 200 hours of backseat QF-4 time by that time too.
I went through the RAG with a flight doc who got his aviator wings and was in Prowlers about 15 years ago, he was the first of his kind in over 10 years and they 'formalized' it after he did it but I don't know of they still have the program. I think he was in Brett's squadron at one point, maybe? Heard they had a hard time figuring out what to do with him.
There are also the guys who are aviators then go to med school afterwards, some of them continue to fly like the doc in my last squadron who was a former B/N, I think he did a shortened ECMO course and definitely flew more often than the normal flight doc.