Sauté onion in butter until tender. Combine onion, liver, breadcrumbs, eggs, flour, and salt in a medium mixing bowl; stir until well blended.
stir until well blended. Bring broth to a boil in a small Dutch oven. Drop liver mixture by teaspoonfuls into boiling broth. Reduce heat; cover and simmer 30 minutes. Remove dumplings with a slotted spoon to a warm platter; discard cooking liquid. Serve immediately.