I suspect one reason why you don't see recipes is that the breasts are generally cooked separately. Ariane Daguin of
D'Artagnan and Paula Wolfert, author of
The Cooking of Southwest France
talk about treating the breast as if it were a steak -- just searing it and serving it with frites.
But I confit duck breasts (on the bone, with skin) along with the legs. Comes out very nice -- they cook in a shorter time, so you have to remove them before the legs are done. In fact, I've done almost all parts of cut-up ducks, including the hearts, gizzards, wings, and backs. You can't serve those as an entree the way you can the leg or breast, but they supply good meat for salads and garnishes.
As for beef, I guess you could do it in a similar way. The main difference is that you'd have to store it in meal-size portions. The fat will harden completely, and if you stored a large quantity of the meat, you'd have to warm all of it before you could take out just as much as you needed. That is not good food safety.
Then again, maybe you could do something like carnitas with beef, or like rillettes (although it would not be soft and unctuous the way the pork and duck versions are).
ETA: Going back to the idea of regular old braised short ribs: once the meat is cooked, there are lots of things you can do with it. I add different spices to the basic sauce for different flavors (Indian spices one time; Jamaican another). I dice or shred the meat to use in pasta sauce, or to make something like ropa vieja. And I chop the meat very fine to use as filling for ravioli.
