I became coeliac with pregnancy but I remember vividly what a great sandwich tastes like and if you ever come to Florence you must make time for one of the best sandwiches in the world from All'Antico Vinaio.
Vegetarians look away because this is the place to stuff your schiacciata (Florentine for focaccia) with sbriciolona (a type of salami), finocchiona (another type of salami with anise seeds), prosciutto and salame Toscano topped with all sort of grilled vegetables, mozzarella, pecorino and sauces such as truffle cream and sun dried tomatoes spread. I mean, a vegetarian could have a great sandwich here too.... but you would sort of be missing out.
Don't expect anything fancy from the experience - the sandwich shops are two simple shops off Piazza della Signoria: one where you can sit down and one where you can only to take-away. You will probably end up sitting by the Uffizzi Gallery or by the Arno overlooking Ponte Vecchio to enjoy your sandwich - which are both not bad at all eating spots. Be prepared to cue at pretty much any time of the day, but these run pretty quickly and ultimately are so worth it!
If you prefer to sit down and have a more "fine dining" experinece, you can grab a table at the less busy Osteria next door - no sandwiches served here but you can enjoy primi (risotto, pasta, soups...) secondi (meet and vegetables) or the phenomenal taglieri (boards) which you can get in all sizes and variations with a side of foccaccia and a glass of red wine. You can book here at any time and it's a great spot for both lunch and dinner.
All' Antico Vinaio
Via Dei Neri 74 R, 50100 - Firenze