In American Hustle Irving said his friend Ed Malone is Cold Cut King of Long Island. What does that mean?
I'm not sure which part of the phrase is confusing, but I'll give it a shot.
A "cold cut" is another name for a thin slice of cold processed meats or other cooked meats. Examples include thin-cut ham or baloney.
If someone is the "cold cut king", then presumably that person either sells a lot of cold cuts (slices of cooked or processed meats), or eats a lot of cold cuts.
Naming a location implies that the person is known throughout that area for eating or selling a lot of cold cuts.