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Beat the egg whites until they hold firm peaks, then fold in the egg yolks, noodles, cheese, and cilantro. Season with a pinch or two of salt. Heat 1/4 inch of oil in a medium skillet. When it's hot, drop in the batter, dividing it into 4 portions by eye. Fry until golden, then turn and fry the second side, about 2 minutes in all. Drain on paper towels until needed.
Warm the olive oil in a soup pot. Add the leeks and potato and cook over moderately-high heat, stirring occasionally, for 5 minutes and add 1/2 cup water so that the vegetables stew rather than fry and cook for about 10 minutes. When the potato starts to soften, add the chard, cilantro, and 1 teaspoon salt. Cover the pan and cook until the chard has wilted after 5 minutes or so, then add the 6 cups of water or stock. Bring to a boil, lower the heat and simmer, partially covered, until the potato is soft, about 20 minutes. Taste for salt and season with pepper. Ladle the soup into soup plates, add a noodle cake to each plate, and serve immediately.