I wasn’t familiar with both of these expressions but I looked it up and “a la mode” is an American culinary expression, meaning “served with ice cream”. And “au jus” is also an American culinary expression, meaning “gravy” or “broth”. Now, even though they are both derived from a French expression that is a prepositional phrase with à (meaning with), it does not matter any more when they were borrowed to English.
“A la mode” became a new adverbial expression meaning just that: “served with ice cream”. You can have pie a la mode = pie served with ice cream, but obviously not *pie with a la mode = pie with served with ice cream.
And “au jus” became a noun expression meaning “broth” or “gravy”. And you must say sandwich with au jus = sandwich with gravy and can’t say *sandwich au jus = sandwich gravy.
What is extremely interesting here is that it bothers the prescriptivist who wants language to be a certain way he feels it is supposed to be, also the author on that webpage.