Is "swimming in broth" a common cooking technique for pasta and broccoli rabe?

I am not sure whether it is a common way to prepare it in the US, but I would say that it is definitely not the traditional Italian way to serve pasta salsiccia e friarielli which is instead a fairly dry pasta dish.

Of course nothing stops you from making variations on the theme. For instance, I know some people who purée the broccoli rabe.

However, in Napoli (where the dish is from) you would generally have it dry.

The fact that the rabe and pasta were overcooked (they bring shame upon my home town!) would make me question the authenticity of anything they do in those restaurants. The answer as to whether you add stock or water will vary in Italy from region to region, and town to town. You'll never get a bigger argument with Italians from diverse regions then asking them how pasta carbonara should be made (dry or wet), and each will have their own "right" answer. In the north it's probably wet where the eggs are added to the pasta after it has been removed from the water, in the south it's probably dry with the eggs cooked in the pan and the pasta added after. What's right in this case depends on how you personally like it because you are in the enviable position of being able to cherry-pick what you like out of any cuisine.

So the answer is that there is no right answer as there are many different regional styles of Italian food, and it all comes down to personal taste. I make this dish myself, and I usually do not use stock, what I may do is add a ladle of pasta water because I like having a bit of liquid. Try it both ways, see how you like it.

It appears that some recipes do call for using a small amount of stock. Perhaps this restaurant just used an excess.

http://www.examiner.com/article/penne-pasta-with-sausage-and-broccoli-rabe-recipe