Me and a friend are arguing on which is correct. Which one would be more correct and why?
When cholera did break out in Britain Farr believed, like Chadwick, that it was caused by miasma.
or
When cholera did break out in Britain, Farr believed, like Chadwick, that it was caused by miasma.
The comma you add is not required; the author probably felt that putting it in would make the sentence choppy and harder to follow.
But I think the comma is advisable, to make the syntax clearer. I would rather eliminate the commas bracketing like Chadwick, which is a very light phrase which does not seriously obscure the syntax. Alternatively, like Chadwick could be bracketed in parentheses.
I would write the sentence as follows, just to avoid that Britain Farr is parsed as single noun. (When the reader reaches believed, it is clear that Farr is the subject of believed, though.)
When cholera did break out in Britain, Farr believed (like Chadwick) that it was caused by miasma.
In the same way, I would write the following sentence with a comma (but for other reasons).
When you see him, give him this note.
I would use the extra comma, based on this guidance found in the Purdue OWL:
Use commas after introductory a) clauses, b) phrases, or c) words that come before the main clause.
So, for the purposes of determining whether or not there should be a comma, let's rephrase your sentence:
When cholera did break out in Britain, something happened.
As the Purdue OWL goes on to say, “introductory clauses are dependent clauses that provide background information or "set the stage" for the main part of the sentence, the independent clause.” In this case, the main phrase is the part we just shortened to “something happened”:
Farr believed that it was caused by miasma.
I would follow the OWL's guidance, and put a comma at the end of the introductory clause that starts your sentence.
