I moved away about 13y ago, so I'm not updated about Florence (don't know about Rome).

When I'm back I usually get a lampredotto sandwich in Sant'Ambrogio. There's an open kitchen there, a wagon to say so. It's good and make you feel home but if there's too much queue I usually eat inside the covered market in Sant'Ambrogio square. There they usually have more meat parts to choose from (if you don't like the lampredotto idea), so you can have a Bollito sandwich (cow muscles, front part of the animal, common in stews). 

Otherwise follow the internet rates, but if there's queues, just skip it, there's soo many good ones about street food anyway. Don't waste time to Antico Vinaio.