Place the ribs in a large pot and cover with water. Bring to a boil; reduce heat to medium-low, cover, and simmer until the meat is falling from the bones, about 2 hours. Remove the ribs and allow to cool. Trim the meat from the ribs and cut into bite sized pieces; refrigerate. Continue simmering the broth until reduced to 1 cup; refrigerate until the fat congeals to the top.
Skim and discard the congealed fat. Pour the broth into a saucepan along with the meat trimmings, beef stock, potato, mushrooms, salt, and pepper. Bring to a boil; reduce heat to medium-low, cover, and simmer until the potatoes are tender, about 20 minutes. Sprinkle with chives to serve.
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