Oh bother
Whether you call them dictionaries or grammars, 'dialects' have been studied and documented for many a decade, and in some cases for a couple of centuries now.
As far as 'language' vs. 'dialect'...it depends on whose definition you're talking about. In the linguistics realm, the distinction is pretty clear: you have 2 'dialects' when there is some level of mutual intelligibility--albeit strained, as I can think of many friends who couldn't watch "Trainspotting" without the English subtitles, although I didn't need help in understanding (most) anyone. On the other hand, there are 2 'languages' when there is zero intelligibility.
What most people use as the criterion for 'language' vs. 'dialect' usually centers around political lines: there is usually only "Chinese" (until you realize that Cantonese, Mandarin and the like are very different, and no where near mutually intelligible!), "Serbian" and "Croatian" are separated (yet there is a very high level of mutual intelligibility...yet you will never hear either a Serb or a Croat tell that story!), as well as "Norwegian" and "Swedish" are traditionally separated (again, they're usually considered dialects of each other).
The Italians consider the different regional 'speeches' to be dialects, although I will tell you from personal as well as other antectdotal accounts that I cannot understand most Southern Italian speech (Sicilian in particular), so one could argue with that. I would argue, based on the linguistic data, that there might be a 'Sicilian language,' because of actual differences in the inflectional structure. But there is much debate regarding this, and I really don't have the time nor space to do it justice.
As for the dessert...I thought Neopolitans just had that funny ice cream