Look for thick pieces of salt cod when you shop, they are easier to shred and are better quality than thinner pieces. Slideshow: More Latin American Recipes
In a large bowl, cover the salt cod with cold water, refrigerate and let soak for 2 days, changing the water every 8 hours. Drain.
In a medium pot, cover the salt cod with water and boil for 15 minutes. Remove the fish from the pot, reserving the cooking liquid. Let the salt cod cool till easy to handle, then use your fingers to finely shred it into a large bowl.
Meanwhile, boil the potatoes in the salt cod cooking water until tender, about 20 minutes. Let the potatoes cool then pass them through a potato ricer into the bowl with the salt cod. Stir in the onion, garlic, cilantro and egg yolks. Season to taste with salt and pepper, then sprinkle the flour over the salt cod mixture and fold it in. Beat the egg whites to soft peaks then fold the egg whites into the salt cod. Form the batter into 10 balls, about 1/4 cup each, then roll them in the bread crumbs.
Line a baking sheet with paper towels. In a medium pot, heat the oil to 350°. Fry the cod balls in batches until golden brown, about 6 minutes per batch. Drain the cod balls on the paper towel–lined baking sheet, then serve.