Self-sustaining meat bird? You bet.
Pekin duck. Excellent feed conversion and fast growth, nice temperaments. Silver Appleyard, excellent feed conversion, fast growth, large size, excellent egg layers, nice calm temperaments.
Any breed of large geese. The geese can be raised on grass since that is their preferred diet. You only need a small amount of supplemental feed and probably would have to feed them in the winter. They don't eat much, considering the size of them.
For a self-sustaining meat chicken, I'd probably go with the Dorking. They are a meat bird, not a dual purpose bird, and their egg laying is a bit scant compared to some other breeds.