Best Fried Artichokes in Rome
You should be asking for carciofi alla giudia, as simply "fried" can be ambiguous. Deep-fried artichoke wedges are another, less well-known, Roman specialty, and very good.
I recently had carciofi alla giudia at Piperno and Giggetto and thought Piperno's was much better. In any case, with the exception of Ristorante Paris in Trastevere (whose owners worked at Piperno for decades before opening their own place), the Ghetto is the best area to try them. It used to be the only place you could find them, but now restaurants are making them all over town. Sora Margherita's is good too (though I don't really care for the place otherwise). I haven't been to Pompiere in years because of some seriously disappointing meals, but they should be reliable for the artichokes too.
You should definitely order them in the Jewish Ghetto, and here are a few places:
Sora Margherita: This hole in the wall place does them a bit differently than other places. They pre cook them in water, drain them, and then fry them. They become deep golden and extremely crispy. I love them.
Gigetto: They do their's with the traditional double frying method and they are very good. They are at their best in the late spring, when local artichokes are in season. The quality changes throughout the year depending on where they are getting their artichokes from (France, Sicily, Sardegna) but are always pretty great.
Piperno: the oldest and most famous restaurant in the Ghetto, it is the most formal of the bunch. Very elegant dining room, white coated waiters, etc.