break his hand over

There is NO entry for "break sth. over" used in this context anywhere online.

The context is Arthur Conan Doyle's "The Final Problem".

Watson notices Holmes's bleeding knuckles and asks what happened. Holmes replies:

"It is not an airy nothing, you see. On the contrary, it is solid enough for a man to break his hand over."

A bit later in the same conversation, Holmes explains the cause of the injury in more detail. He describes several attacks he suffered that day at the hands of Professor Moriarty's agents, ending with this one:

Now I have come round to you, and on my way I was attacked by a rough with a bludgeon. I knocked him down, and the police have him in custody; but I can tell you with the most absolute confidence that no possible connection will ever be traced between the gentleman upon whose front teeth I have barked my knuckles and the retiring mathematical coach.

So the injury came from Holmes punching an attacker's teeth hard enough to scrape and bruise his own knuckles.

My question is about the phrase "break his hand over" from the first quote.

Does it mean something like "solid enough to injure one's hand by hitting it," similar to saying a punch was hard enough to hurt your own hand in the process?

Is this a common idiom, or an unusual or archaic construction specific to Victorian English?

Any insight into the grammar and usage here would be appreciated.

Over is used as a preposition here, in sense #7 at the Merriam Webster link:

used as a function word to indicate the object of an expressed or implied occupation, activity, or concern

The object of the preposition is the "solid" matter which Holmes is talking about, which has necessitated breaking his hand. Consider other examples and how they could be rewritten in the more common order of subject-verb-preposition-object.

It's not something I'll lose sleep over.

I will not lose sleep over this.

Grammar is something he'd get in a fight over.

He'd get in a fight over grammar.

It is solid enough for a man to break his hand over.

A man could break his hand over this solid matter.

There's nothing particular about the phrasing indicating how the hand was broken by hitting it, although it becomes clear from the context that's what happened. The phrasing just indicates that the hand was broken in some way, specifically in the course of investigating the matter.