One address which is an absolute must if you are visiting Rome is Roscioli ( Via dei Giubbonari, 21,Rome ,tel. 066 864045 ). Initially it was a bakery, later a gourmet shop and a restaurant was added. The products are of unbelievable quality but it's not only them that make the place favorite among chefs and Romans. Roscioli food is autentic ,but at the same time has a modern twist.
Mozzarella di Bufala ricotta
Here you will might have perfectly prepared carbonara or all'amatriciana, but also more contemporary cretions like mozzarella hamburger or soufflé with Parmigiano Reggiano.
The main difference from other all'amatriciana pasta i've had in Rome that here the pieces or bacon were very crispy.
Warm octopus and potatoes salad.
P.S. Thanks again for the address Andrea and Leo!