What does “cheffed-up” in “Traditional ramen that hasn’t been cheffed-up” mean?

It's a attempt to shorten the phrase "cooked in some fancy way that a chef might offer in a restaurant, rather than just boiled as a basic dish."

No, it isn't a common term -- I'd call it a neologism, or possibly a term used only by restaurant reviewers. But the "cheffed-up" formulation would probably be understood in this context by most native speakers, by analogy with "prettied up", "tidied up", "polished up" and other common phrases that use "-ed up" to mean "made more so".

Newspaper writers do have a long tradition of writing in a compressed "telegraphic" style, in an attempt to pack as much information as possible into as little space as possible. That does tend to produce phrases which are hard to understand until you have some experience with this style. Don't assume that the New York Times, or any other newspaper, is a good example of formal English.

The main meaning is "[past tense of verb] up". So you take "chef", turn it into a verb, make it past tense (cheffed), and add "up" and you get "cheffed up". That means that someone has done chef-style stuff to the ramen.

However, and this is pure speculation, there could be a pun involved.

Eff is a euphemistic way of using the word "fuck", based on the first letter of "fuck" being pronounced "eff". "Effed up" is the same as "fucked up" or "messed up" (as in made bad or not how it ought to be). The phrase "cheffed up" would have "effed up" within it, so maybe the author was voicing disapproval at doing chef-style stuff to the ramen.