Make the broth: In a large saucepan, combine the bones with three-quarters each of the celery, carrots, celery root, turnip, and onion. Add the 1 tablespoon salt, whole peppercorns, garlic, parsley bunch, bay leaf, tomato, and 10 cups cold water to the pan and bring to a boil. Reduce the heat to maintain a simmer and cook, undisturbed, until the broth is richly flavored and reduced, about 3 hours.
Remove the broth from the heat and let cool to room temperature. Using a spoon, skim off and discard any scum on the surface of the broth. Pour the broth through a cheesecloth-lined strainer set over a large bowl and discard the bones and solids. Refrigerate the broth until chilled, at least 4 hours. Once chilled, remove the solidified fat on the surface of the broth and discard.
Make the dumplings: In a food processor, pulse the bread until it is finely ground into bread crumbs. Transfer the bread crumbs to a large bowl and add the livers to the food processor. Pulse the livers until finely chopped and then return bread crumbs to the food processor along with the parsley, butter, and egg yolks and pulse until evenly combined. In the large bowl, whisk the egg whites until they form stiff peaks. Scrape the liver mixture into the egg whites, season with salt and pepper, and fold until evenly combined. Using a tablepoon-size measuring spoon, portion and roll the liver mixture into small 1-inch balls and place on a parchment paper—lined baking sheet. Refrigerate the dumpling balls until ready to serve, at least 1 hour.
To serve, pour the chilled broth in a large saucepan and bring to a boil over high heat. Reduce the heat to maintain a gentle simmer and stir in the reserved one-quarter each of the celery, carrots, celery root, turnip, and onion, plus the saffron. Gently drop the dumplings into the simmering broth and cook until they float to the top and are just tender, 6 to 8 minutes. Season the broth with salt and pepper and ladle the broth, dumplings, and vegetables into serving bowls. Garnish with parsley leaves before serving.