Prepare the Chicken:
Shred the rotisserie chicken, discarding the skin and bones. Set aside
Make the Broth:
In a large soup pot, add onions, carrot, celery, garlic, oregano, peppercorns, lemon wedge, parsley, and thyme (optional).
Simmer for 30 minutes.
Strain the soup through a fine meshed sieve and return to a clean stock pot
Reserve the carrots and dice into small cubes.Discard the other vegetables
Add in the chicken, orzo, and carrots and simmer 20 minutes to allow the orzo to cook through.
Prepare the Avgolemono Sauce:
In a mixing bowl, whisk together the eggs and lemon juice until well combined.
Temper the Eggs:
Carefully ladle a cup of hot broth from the soup into the egg-lemon mixture, whisking continuously. This tempers the eggs, preventing them from curdling when added to the hot soup.
Add Avgolemono to Soup:
Gradually pour the egg-lemon mixture back into the soup, stirring constantly. This adds a creamy texture to the broth.
Season and Garnish:
Season the soup with salt and pepper to taste. If desired, add fresh dill for garnish.
Serve:
Ladle the Avgolemono Soup into bowls and serve immediately.