Season the chicken with ½ teaspoon salt and ¼ teaspoon black pepper.
Heat 2 tablespoons of the olive oil in a Dutch oven or large pot over medium heat. Add the chicken and cook until lightly golden brown, 2 minutes on each side. Set the chicken on a plate.
Add the remaining 1 tablespoon oil to the pot along with the carrots, onion, and garlic. Season with 1 teaspoon salt and a few twists of pepper and cook, stirring occasionally, until the onions begin to brown, 5 to 7 minutes.
Stir in the cauliflower rice (but not the orzo if using), dill, parsley, and oregano. Add the chicken broth and return the chicken to the pot. Bring the soup to a boil, cover, and reduce the heat to low and simmer for 15 minutes or pressure cook for 10 minutes.
Check that the chicken is fully cooked and remove it from the pot. If using orzo, add to the soup and continue simmering until the orzo is al dente, 7 to 9 minutes.
When the chicken is cool enough to handle, shred the chicken and return it to the pot.
In a medium bowl, whisk the egg yolks. Remove 1 cup of hot broth and slowly pour it into the egg yolks while whisking vigorously.
Whisk this warmed egg mixture into the soup pot. Stir in the lemon juice and adjust to taste with salt and pepper.
Serve garnished with fresh parsley and dill and drizzle with a little olive oil.
