Detective Comics #77 (July, 1943)

"The Crime Clinic"
Writer: Bill Finger
Pencils: Bob Kane
Inks: George Roussos
Synopsis: Brilliant, fashionable, rich person's surgeon Matthew Thorne leads a double life - he secretly operates a "crime clinic" where instead of examining patients and prescribing medication he examines criminal plans and prescribes methods for carrying them out! Of course, "secretly" is a relative word, since he gets the word out about his services through advertising to the crooks - and also insists he take their fingerprints on file to ensure they aren't cops, which... just... I mean, that's where I'd just walk out if I was a Gotham gangster -- but no one thinks to ask "How do I know you're not a cop, Doc?"
Anyways, Thorne gets a 25% cut of the loot as a consulting fee, and a 50% operating fee if he has to come out on the job himself (where he wears full surgical gear for reasons that kinda make sense, but are mostly because he's a Classic Batman Theme Villain).  
However, on one particular evening when the doctor is operating on a rubber warehouse (rubber being a valued commodity during wartime), the operation is spotted by the Batman and Robin! And so we get a standard pun and props filled fight scene in the rubber warehouse, which apparently contains only manufactured rubber toys instead of any kind of valuable base rubber. While the crooks get the drop on the heroes, Thorne refuses to allow them to kill the Dynamic Duo, because as a Doctor he is sworn to "do no harm" after all. The Doctor escapes, but not before Batman manages to place a "tiny, low power, short wave transmitter" that they can trace with the Batmobile's "direction finder" -- which I believe makes this the first use of a "Bat-tracer", unless I am mistaken.
They follow the Crime Doctor to Matthew Thorne's office and thus realize his true identity. Bursting into his office, we get a fight filled with medical props, including a cool panel where Batman grapples with Thorne behind an x-ray screen. Then Thorne pulls a gun on Batman, because he's decided now that he'll kill if he's forced to. 
But it doesn't come to that, because one of Thorne's legitimate patients bursts into the room with acute appendicitis! He has to be operated on now! So Thorne enlists the help of Batman in an impromtu surgery, saving the man's life! 
After the patient has left, Batman questions the exhausted Thorne -- why does a brilliant surgeon turn to crime? And, if a criminal, why bother to save a man's life instead of using the opportunity to escape? Thorne explains that while he is a doctor and dedicated to saving men, he can't help but enjoy acting criminally. It's a compulsion, he cannot help himself. In light of this admission of madness, Batman declares Thorne his strangest foe!
Then Thorne throws ether in Robin's face and ties up the heroes with rubber hoses. Thorne tells Batman he can't help how he acts, and gives the clue that he's off to look for the Philosopher's Stone. Batman uses the Doc's discarded cigarette and ether bottle to make a flame to heat up the rubber hoses and thus cause them to expand and allow him and Robin to escape.
Batman explains that the Philosopher's Stone was an old myth of a substance that could turn base metals into gold, and so Thorne must be going after a physics professor who has a formula with which Thorne believes he can use an atom smasher to change the atomic order of objects to make gold!
So Thorne steals the formula and heads off to the Great Eastington Atom Smasher, with Batman and Robin hot on his heels. Now, for translation from nineteen-fortiesese, an atom smasher is a particle accelerator, and Finger is probably referencing the cyclotrons and calutrons that were being built to produce materials for the Manhatten Project, not that Finger would know that or what they were for beyond a vague knowledge of the basics of popular understanding of atomic science.
Kane and Roussos also demonstrate that they have no idea what an atom smasher looks like, or else didn't care, drawing it as a comically large metal balloon-like structure protruding out the top of a small little box-like building, with some stairs and ladders nonsensically going up the side to the top (why would anyone need to go up there?)
Once inside the building, we can see that the visual inspiration is the Crockcroft-Walton generators used in nuclear disintegration, but we don't spend much time inside as soon the characters are chasing on those ladders going up the outside of the building to the top, and it's clear that the whole purpose of this was to enable the characters to battle dramatically in a high place.
Thorne announces his intention to commit his first murder and kill the Batman, but Batman punches him off the top of the atom smasher and Thorne falls into the convenient river just below the drop. Showing more foresight than he ever has before in this situation, Batman dives into the river to ensure Thorne's capture and so the Crime Doctor is arrested (but is promised a return in Batman #18!)
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My Thoughts: The Crime Doctor is a Batman villain who has lasted through to the modern day, but still never managed to become anything more than a footnote character. His lasting claim to notoriety happened when he was retconned into being the brother of the much newer but more popular gangster character Rupert Thorne. 
In modern interpretations Matthew Thorne is usually depicted one of two ways - either as a psychotic mudererous surgeon, the medical serial killer gruesomely doing away with his patients; or as a "crime doctor" in the more common sense of the term, which is a corrupt doctor who works for criminals to patch them up from wounds since they obviously cannot go to hospitals. It is in this second sense that he is usually connected to his mobster brother Rupert.
However Finger chooses neither of these, and in terms of actions he makes the Crime Doctor fairly standard -- what makes the story special is the way Thorne is depicted in terms of characterization. The Crime Doctor feels like a true member of the Rogues Gallery not just because he's a "theme" villain, but because of the unique focus Finger gives on his psychology. This focus is what has defined, and continues to define, the best of Batman's villains.
The Art: So the way things normally work is Kane does some flat, undefined pencils and then Roussos or Robinson comes in and give definition and texture to Kane's characters with inks and copious shadows. In this case, I feel like Roussos almost goes overboard. There are some really great dramatic panels and great noir-esque "lighting" but it really does go too far in some panels where Roussos has just drown the characters in ink to the point where I feel like he's just covering up Kane's flat art more than trying on purpose to be dramatic. That being said, several of the panels are very dynamic and cool looking, even if the design of the atom smasher building for the climax does look patently ridiculous.
The Story: In all honesty, the story in this issue is perfunctory. The whole idea of the Crime Doctor as a consulting criminal devising plots for lesser gangsters for a fee is one that has already been used with The Joker and The Penguin in the past and even goes back to Professor Moriarty. The plot with the Atom Smasher comes out of nowhere and doesn't seem motivated at all with what Thorne had been doing. No, what makes this issue golden is that none of this matters -- the jewel of the story is the centre section where Thorne's psychology is examined. The idea that he is compelled, that he can't help himself, that he simply feels good committing crimes, that it excites him, but that otherwise he's a moral, brilliant surgeon. Only even that goes down the drain -- we actually see Thorne's psychological unravelling over the course of the story, the way that his criminal impulses corrupt him: At the start of the tale he refuses to kill, midway through he'll kill if he needs to, and by the end he's blatantly murderous. It's a great series of subtle character details that give Thorne an arc, and show that Finger really put some effort into the characterization and psychology of his villains above and beyond the norm for Golden Age storytelling. The last time he was this good, however, was probably the Two-Face two-parter.
Notes and Trivia: First appearance of Matthew Thorne, the Crime Doctor and first appearance of Bat-tracers.