Sprinkle liver with salt and pepper; dredge in 1/3 cup flour.
Melt shortening in a large skillet over medium heat; brown liver on both sides. Remove and keep warm, reserving pan drippings in skillet.
Sauté onion in pan drippings until tender. Stir in 1/4 cup flour; cook 1 to 2 minutes. Gradually add water, stirring constantly, until mixture is thickened and bubbly. Return liver to skillet. Cover and simmer 5 minutes.
Prepare dumplings. Drop dumpling batter by tablespoonfuls into liver and gravy mixture. Cover and simmer 5 minutes. Turn dumplings; cover and simmer 5 additional minutes. Serve immediately.