Roman Suckling Lamb (Abbacchio alla Romana) in Rome
My guess is that you mean "Abbacchio al Forno". As mentioned, there's "al scottadito" as well, but that's a grilled dish.
I've had reasonable luck at Da Felice (people say it's not the same as it was, but it's still OK) and Matricianella (which you really MUST book for if you hope to get in) Matricianella is a lot more central, at this point perhaps more atmospheric, and they have a better selection of other Roman classics. The potatoes at Matricianella (the "canonical" accompaniment) were much better than current-version Felice.
I suspect it's possible to do better than these, but so far I've not found anywhere dramatically better, and it's not been for lack of trying.
baby lamb is pretty common on Roman menus- we've had delicious versions at Armando al Pantheon, and other places. Armando has the "scottadito" version but we have also had other variations - their online menu is not complete.
Here is a pic of the dish from Katie Parla's blog, but I have to say that these pieces look bigger than some we have encountered
I am unaware of a dish called abbacchio alla romana. Roman restaurants don't go in for slow cooking in a big way, with the exception of coda alla vaccinara (stewed ox tail). Milk-fed lamb is, of course, typical, but the usual ways you find it is as grilled rib chops (abbacchio scottadito) or roasted with rosemary and potatoes (arrosto), which is probably what your source is referring to. Less common, but certainly traditional, is abbacchio alla cacciatora, stewed with rosemary and vinegar. Abbacchio arrosto can be delicious, but it's sort of hit-or-miss in restaurants since it can be dry, and the same restaurant can serve it perfectly or on the dry side, so you never know. However, Al Moro, Nerone, and Felice are three places that almost always have it on the menu. There are undoubtedly many others, invariably of traditional type. Whatever you do, don't miss the potatoes. For scottadito or cacciatora, Checchino dal 1887 is good, but they don't do arrosto. Another very traditional way to do lamb is abbacchio brodettato, with a sort of egg-lemon sauce. It requires a certain skill and attention to make, so it has practically disappeared from the trattoria menu, but you might find it in May, though around Easter would be more likely. Checchino has made it in living memory, but it's not on the regular menu.
This definition comes closest to what Maureen describes as "alla cacciatora", but leaves space for the other mentioned preparations, too. I would not call any of them "slow cooked", maybe that is causing confusion. Cucchiaio d'Argento and Mario Batali have dishes they call abbacchio alla romana that are with vinegar and not slow cooked. Some sources describe the "cacciatora" as with an anchovies sauce, where as there are also that use the anchovies under the "romana" name...
In any case agree with Maureen's recommendations as possible venues, adding that Cesare al casaletto has sometimes great lamb preparations, too, esp the scottadito.
Since neither Mario nor Cucchiao d'Argento is Roman, perhaps that's a term used in other parts of Italy to describe a dish associated with Rome. what you describe is cacciatora. The article in the link uses the term for the meat, not the dish. Lamb isn't' served with anchovy sauce, but a single anchovy fillet is used in cacciatora. I'd like to know what OP was quoting. "Abbacchio alla romana" would be a very unusual menu item in Rome.