Gabriele Bonci's famous pizza-by-the-slice joint serves some of the best pizza in town. The slow-leavened dough is made from organic stoneground flour, and toppings change throughout the day. Pizzarium also sells excellent bread and supplì (fried rice balls with various fillings). If you dare, get one of every slice. Doing so will certainly push Pizzarium out of the moderate budget range, but it is a worthwhile splurge. Pair your pizza with a craft beer from the fridge. Beware: the tiny place gets crowded at lunch, there are only a couple of benches outside to sit on, and there is no table service.
If you happen to be near the Vatican around lunch time you MUST visit Pizzarium! It might seem a bit out of the way but it is well worth the walk. The pizza is cut by a scissor based on how hungry you are!