The preposition a in food phrases

This is something I'd like to clear up because it irks me. I've seen it a number of times. In the names of Italian dishes, the preposition a is often found, in its compound form (preposition plus article.) Now I have read many times (unfortunately, by my fellow Americans, although I suspect that all English speakers tend to make this mistake) that such phrases in the feminine singular refer to a woman or a wife. Examples:

gnocchi alla romana

sogliola alla mugnaia

spaghetti alla carbonara

pesto alla genovese

They come up with fanciful explanations such as "coal miner's wife spaghetti" for spaghetti alla carbonara, or "female miller's sole" for sogliola alla mugnaia. No, no, no. These phrases are feminine because a feminine word is intended but not expressed, for brevity. Probably the word moda. So that gnocchi alla romana means "Roman-style gnocchi" and sogliola alla mugnaia means "miller-style sole." French also does this- their equivalent word mode is also feminine.

This is not to be confused with another common use of compound a in food phrases. This is more straightforward, and simply means "with." Yes, I know that a means "at" or "to," but that's just how idiomatic the use of prepositions is. Some examples:

biscotti alle mandorle (almond cookies)

spaghetti alle vongole (spaghetti with clams)

risotto agli asparagi (asparagus risotto)

pizza ai quattro formaggi (pizza with four cheeses)