A surgeon was a specialist in invasive care within medicine. Unlike physicians, surgeons were trained to practice surgery.
Harry Sullivan was a Royal Navy surgeon, (TV: Planet of the Spiders) specifically a Surgeon-Lieutenant. (PROSE: The Face of the Enemy)
Cassandra O'Brien had two surgeons with her in later life who had a responsibility to moisturise her. Their absence led to the skin drying out and tearing violently. (TV: The End of the World)
Dr Grace Holloway, a cardiologist, performed surgery on the Seventh Doctor. (TV: Doctor Who)
To claim the Halassi Androvar from King Hydroflax, River Song hired a surgeon, sending Nardole to collect him. Nardole instead brought back the Twelfth Doctor, who had deemed "surgeon" a "close enough" title to his own, leaving the real surgeon behind. (TV: The Husbands of River Song)
The V-Time Experiment was overseen by a Superior who called themselves "The Surgeon". (PROSE: The V Cwejes)