Add the olive oil to a dutch oven over medium heat. Add the onions, garlic, carrots, and celery and cook until the onions are translucent, about 3 minutes. Add the chicken to the pot followed by the broth and the rice. Season with salt and pepper. Bring the soup to a boil, turn down the heat, cover the pot and simmer for 15-20 minutes, stirring occasionally, until the chicken and rice are both cooked through.
In the meantime, in a small bowl, beat together the egg with the lemon juice. When the soup is done simmering, ladle some of the hot broth into the lemon and egg mixture, stirring constantly to combine (so as not to scramble the eggs), then add the lemon and egg mixture to the pot, stirring until well combined. The soup will thicken slightly and become creamy at this stage. Stir in the fresh dill and generously season the soup with salt and lots of freshly cracked black pepper.
To serve, ladle the soup into bowls and top with fresh dill, chopped parsley, lemon slices, and a drizzle of extra virgin olive oil as desired.