I find it worrying that I don’t remember participating in this thread. Given what I was going through in my life at that particular time, though, I think I acquitted myself reasonably well.
Sorry, but language doesn’t work that way. And your example doesn’t apply since words are adopted into English differently. Once words are adopted into English, they have their own “grammatical integrity” (whatever that means). It may match the other language, or it may match English usage.
Since the meaning is clear, “with au jus” is not incorrect in English simply because it’s a mixture of languages. In this case, it would perfectly describe a dish where the au jus was on the side. “Roast beef au jus” is roast beef served in its own juices; “Roast beef with au jus” is roast beef served plain with the juices on the side (which is generally how it’s served in the US.
A mixture of languages does not make it incorrect, but the annoying thing is that the English usage was adapted by people who want to sound exotic by using a foreign phrase but don’t actually understand what it means. Then the error is proliferated so profusely that it is no longer considered an error. It is exactly parallel to the phrase Colophon gave, “chili with con carne,” except no one actually says that.
Your arguing against a two-year-younger version of me, but I still hold that your examples above sound completely ridiculous.
And I believe I would be correct in saying it is still incorrect in British English, and possibly just-about-acceptable in American English, though verging on the ignorant. FWIW, even Wikipedia agrees with this position.
The noun is “jus”. “Au” is a preposition. As someone said, it’s equivalent to chili with con carne.
I think it’s completely accepted by all but the most persnickety of speakers here. It’s even gotten to the point where I (who speak a smattering of French) have gotten to using phrases like “with au jus gravy” or “would you like some more au jus,” as annoying as it may be.
But I’m sure this can’t be the only example in English where a word’s original meaning had been lost and there is a redundant preposition or article or whatnot in it. While “with con carne” doesn’t exist, I know there’s others, but my brain database search isn’t returning anything.
If you can’t think of any examples, just finish the statement with “and etc.” 
And I think it’s funny that saying “it can only be that person’s head!” is mentioned as support for not using a possessive pronoun. As if “it’s implied” means that we should automatically default to the definite article.
I’d also like to point out that we do say “I got punched in the head” or “a shot in the arm”
Ah, “the hoi polloi” was the one I was thinking of. Also, “please, RSVP” may qualify. “on the a la carte menu” may be a little weaker of an example, but another one that works.
“The al-Qaeda terrorist group.” “The La Brea tar pits”?
“The hoi polloi” is one example. But I’d disagree that either that or “with au jus” is really “completely accepted by all but the most persnickety of speakers” here. Unless you’re willing to admit that there are a lot of persnickety speakers and that many of them (including authors of style guides, etc.) are pretty influential in determining the form of written Standard American English.
I do agree, though, that “with au jus” has become a common enough usage that everybody at least recognizes what it’s supposed to mean. If a waitress told me that the special was “duck with a l’orange”, on the other hand, I’d probably be sufficiently confused to ask her for clarification, although I know what “duck a l’orange” is.
[ETA: jinx, you owe me a Coke!]
American English doesn’t have a concise and precise term for the “jus” of au jus, and since most English speakers won’t instantly recognize the “au” as a preposition, the phrase au jus just readily fell into the role of a noun that indicates the specific type of “juice” typically served with roast beef. Any other way to describe it accurately is cumbersome (too many words). We NEED to use “au jus” as a noun – NEED to, I tell you!
Maybe the preferred form should be “I got punched in head.” and dispense with the article entirely. (It was probably written for the Post anyway. We’re better off without it.) Then we’d have stern, muscular, Hemingway speech, assuming Hemingway was a markedly unsuccessful prizefighter. We’d have the kind of speech Strunk and White advocated, when they yelled “OMIT WORDS!” before smashing down a superfluous brick wall. It’d be the kind of speech used by simple farmers, people of the land, the common clay of the New West.
Perhaps you meant to address that to jjimm?
I did specifically say “speakers” for a reason. I know it is not considered stylistically acceptable by most, if not all, style guides to write “with au jus.” My point is that, in casual conversation, from my observation, American speakers use that construction in such a way that would lead me to believe that it has become assimilated into the English language as a single grammatical unit by most American speakers. Now, I may be wrong about the pervasiveness of it, but given how common it is in menus and even television commercials, in addition to my observations with friends and acquaintances of all educational levels, I’m willing to bet it’s pretty much accepted at this point.
Surely you meant to say “It’d be kind of speech used by simple farmers, people of land, common clay of New West.” 
You mean, idiots?
I hope this isn’t (too much of) a hijack, but I was in a restaurant one time and asked the waitress, “Excuse me, but what is the soup du jour?” while pointing at the “Ask About Our Soup Du Jour” sign in the menu.
She looked at me and replied with a trace of a sneer in her voice, “It means, ‘Soup of the Day’!”
Took all of my self-control not to say anything at all since I didn’t want “soup avec le mouse turds”.
That’s exactly what I was doing, way back then…
Anyway, what type of sauce does “au jus” refer to in America? We don’t tend to use the phrase much in England, either with or without the “with”. Isn’t jus just gravy? Why not call it gravy?
It’s not gravy here in the U.S. It’s thinner, and usually translucent, more like a seasoned beef broth than a gravy (which would typically be opaque and thicker, due to use of flour). Au jus is commonly served with sliced beef dishes, including the staple of diner-style restaurants, the “French dip” sandwich (sliced beef on a French roll, with a cup of au jus, into which one dips the sandwich).
Generally, au jus is not used to describe what would be called gravy. “Gravy” is generally a thickened sauce made from flavor extracted from meat. Au jus is used to refer to something more like broth. Of course, maybe where you are “broth” is used differently. Broth is a thin liquid flavored by meat extracted from pretty good cuts of meat, as opposed to “stock” which is extracted from bits of the beast that are full of flavor, but not as directly edible – bones, membranes, less choice bits of meat, ect.
I’ve never encountered that delicacy. A clear meat broth is also known as a consommé, but I’ve not seen it used as a dip - only as a soup course.
“Gravy” to me can mean anything from thick gloopy dark-brown type made from a packet to the delicate thin translucent stuff. “Jus” only tends to be used on menus that are entirely in French.