Parboil the rice: In a pot, boil 1 cup of rice in 2 cups of salted water or chicken stock. After 5 minutes, drain and allow rice to cool in the refrigerator.
Make the meatballs: In a food processor add the onion, parsley, dill, and two egg whites (save the remaining 3 yolks for later in the recipe). Add pinch of salt and pepper to mix. Process for 1 minute until herbs and onion are finely chopped.
In a large mixing bowl, add the herb mixture, cooled parboiled rice, and the ground beef. Mix well to combine, and refrigerate mixture for 30 minutes.
Form the meatballs and lightly flour them (or roll them in flour). To a large soup pot or Dutch Oven, and add the olive oil and heat on low heat. Add the floured meatballs to the olive oil and cook lightly.
Pour the chicken stock over the meatballs, bring to a boil, then reduce soup to a summer. Allow soup to cook for 30 minutes.
Add the chopped escarole, and cook for another 20 minutes.
In a separate bowl, mix the egg yolks and the lemon juice, and whisk to combine. While whisking, slowly add a few ladles of the hot soup broth to the egg and lemon mixture, until it becomes creamy. Add the egg lemon avgolemono sauce to the soup pot.
Taste, and add salt and pepper if desired. Serve hot with extra dill or parsley to garnish.