In a large pot, heat the olive oil over medium heat. Add the diced onion and garlic, sautéing for 3-4 minutes until softened. Add the chicken breasts and pour in the chicken broth. Bring to a simmer and cook the chicken for 15-20 minutes, or until fully cooked. Remove the chicken, shred it, and set it aside.
Add the rice to the pot and cook for 15 minutes, or until tender.
In a small bowl, whisk together the eggs and lemon juice until smooth. Slowly ladle a cup of the hot broth into the egg mixture, whisking constantly to temper the eggs and avoid curdling.
Gradually stir the lemon-egg mixture back into the pot with the soup. Add the shredded chicken back to the soup, season with salt and black pepper, and let it simmer for a few more minutes to thicken.
Ladle the soup into bowls and garnish with fresh parsley or dill. Serve hot with crusty bread or a side salad.
