It sounds like you're asking for an answer in terms of CGEL terminology. Since you asked on ELL, though, I'll write you an answer targeted at someone learning English rather than learning CGEL. I offer no rules here, only an explanation of what's going on. It's probably not what you're looking for qua linguist, but hopefully you'll find it useful qua learner of English.
People understand chicken soup by analogy with phrases like these:
Each of these phrases functions as a noun, and each consists of two words that are themselves nouns. The second word names the broader kind of thing that the phrase as a whole denotes (the "genus"); the first word names what is distinctive about the the thing that the phrase as a whole denotes (the "differentia"), that is, the essence of what distinguishes it from others of its kind.
So, a reference manual is a manual is distinguished from other kinds of manuals, especially a user's manual, by being designed for reference, not for introduction or teaching. A beauty contest is distinguished from other kinds of contest by the fact that the contestants are judged on their beauty. A football team is distinguished from other kinds of team: basketball teams, baseball teams, hockey teams, etc.
Many compound words are formed in the same way. For example, gumball, landmark, raincoat.
What "one" refers to
Because the two words function together as a single noun, it feels incongruous to break them apart. Usually the pronoun one refers to something of the same kind that has just been referred to; e.g. "Do you want the blue gumball?" "No, I want the yellow one." So, it's more natural to use one to refer to the whole phrase. "Do you want the blue football helmet?" "No, I want the yellow one."
Sometimes, for short, people will say the differentia, omitting the genus. For example: "What kind of soup do you want? We've got chicken, lentil, cabbage, and mulligatawny." That's in lieu of saying "We've got chicken soup, lentil soup, cabbage soup, and mulligatawny soup." Most people wouldn't normally use the word mulligatawny by itself; normally it occurs followed by soup. Mulligatawny soup is perceived as a single noun, just like chicken soup, etc.
You can't make a rule out of that, though, because sometimes the word for the differentia doesn't suggest the genus strongly enough to stand alone. For example, it would sound very strange to say "What kind of machine do you want? ATM or sewing?" Even though both have the same word for the genus, we don't think of them as distinguished from each other.
By the way, "genus" and "differentia" are not commonly used words. Don't expect the average person on the street to know what they mean. I use them here only because I can't think of clear synonyms from everyday language.
If you want a "test", here are two:
(1) Chicken soup doesn't just mean soup with chicken in it. It means that chicken is the main ingredient. It's the kind of soup. If the soup has a lot of salt in it, you call it "salty soup". If you called it "salt soup", that would mean that salt is the main ingredient—that salt defines the soup (which sounds disgusting).
(2) The "chicken soup" construction has a certain rhythm and intonation in speech, which isn't represented in writing. This can't be stated precisely in words, but I can give you an idea of what to listen for. The first word is spoken at a distinctively higher pitch, with some feeling of tension; the second word resolves the melodic tension, falling back to the "tonic" pitch of its context. (The speaker's pitch might need to fall still further by the end of the sentence.)
In "salty soup", the adjective is usually said a little lower or a little higher than the noun, but not as much. A speaker might raise the pitch of "salty" up as high as "chicken" for emphasis—that is, to make the listener treat "salty" momentarily as the defining difference:
"No, I was wondering how they made the salty soup, not the bland soup that neither of us liked."
In this sentence, "bland soup" would also likely get the same kind of intonation as "chicken soup". Ordinarily, it wouldn't.
Naturally, as always with prosody, different people do it differently, it's different in questions, it varies a lot to suit the melodic context, it varies by dialect, and some people have a tin ear and don't produce it or respond to it. But this does give you something of a test. You can use it to tell when an adjective plays the role of differentia—for in fact, the first word in these phrases doesn't have to be a noun.
For example, if you say "legal advisor" with the same rhythm and intonation as "chicken soup", that means someone who gives advice regarding the law. Trying to use "one" to refer to just the "advisor" part without "legal" clashes with the way "legal advisor" is heard as an indivisible noun. So, "I already have a legal advisor, can you introduce me to a financial one?" sounds strange. But if you say "legal advisor" with the same rhythm and intonation as "salty soup", it means an advisor whom it is legal for you to hire or consult, and "one" can easily refer to just the "advisor" part:
"Kushner's relationship with the President means he's not a legal advisor to hire, but if the President wants an illegal one, who's going to stop him?"
Here's another example. If you say flying machine like "chicken soup", that means an airplane or another motorized airborne transport, like a dirigible. If you say "flying machine" like "salty soup", that probably means a machine that someone has hurled into the air.
Still, none of this is a rule. In the English language, the word "English" is certainly the differentia, but it gets the rhythm and intonation of the salty soup.
More than two words
Burger King is the name of an American fast-food restaurant chain. The word Burger indicates the kind of king: a king of burgers, as opposed to a king of comedy, a king of railroads, etc. Even a proper noun can work this way. Now notice fast-food restaurant chain. This extends the analogy of the two-word phrases above to more words. In the last fifty years or so, people have been making longer and longer "noun pile-ups".
Notice the usual pattern at work here: What kind of chain? A restaurant chain. What kind of restaurant chain? A fast-food restaurant chain. Notice that the noun phrase "fast food", consisting of an adjective modifying a noun, functions as a noun modifying "restaurant chain".
Notice also that a regular adjective, American, can modify the whole "noun cluster". It would seem incongruous to insert American into the middle of fast-food restaurant chain, because the latter is perceived as a compound noun.
Big noun pile-ups are especially common in bureaucratic writing. I'm only making this one up, but it's typical:
The Fort Collins Real Estate Oversight Committee Chairman
Notice that Fort Collins and Real Estate are themselves two-word nouns. When so many nouns are piled up like this, the result is hard to parse, even for native speakers. The pile-ups in newspaper headlines are often even more bizarre and harder to parse.
Finally, notice that noun cluster and noun pile-up, the everyday names for these things, are themselves instances of what they name.