Heat the 1/4 cup olive oil in a large pot over high heat. Add the onion, 1 teaspoon salt, and pepper to taste; cook until slightly softened, about 5 minutes. Add the chicken, the broth, 4 cups water and the juice of 1 lemon; cover and bring to a boil. Reduce the heat to medium, add the pastina and simmer until the pasta is cooked and the soup thickens slightly, about 15 minutes. Remove the chicken; when cool enough to handle, pull the meat off the bone and shred into bite-size pieces.
Remove the soup from the heat. Whisk the juice of the remaining lemon with the whole eggs and yolks in a medium bowl until frothy. Gradually whisk a ladleful of the hot soup into the egg mixture, then stir the warm egg mixture into the soup and return to medium-low heat. Cook until creamy, about 1 minute. Stir in the shredded chicken, spinach and dill, and season with salt and pepper. Ladle the soup into bowls; garnish with feta, if desired.
Per serving: Calories 425; Fat 27 g (Saturated 6 g); Cholesterol 277 mg; Sodium 858 mg; Carbohydrate 18 g; Fiber 1 g; Protein 27 g