Or that pannini is already plural, just as spaghetti is plural?
In Italian, certainly. In English, no. No native English speaker would say, "There are left-over spaghetti in the refrigerator."
However, English speakers don't say, "There are left-over spaghettis in the refrigerator," (not valid in New Jersey) but they will say there are "panninis" in the refrigerator.
What's wrong with panninis? If it exists, pannini is an English word. The Italian for sandwiches would be "panini."
What's wrong with New Jersey residents saying there are spaghettis in the refrigerator? No one in New Jersey would put
their panninis in the refrigerator because they eat them warm!! What's the matter with youse guys, anyhow??