Sear the Chicken: In the large pot, heat one tablespoon of the olive oil over medium-high heat. Pat the chicken breasts dry and season them generously with salt and pepper. Place them in the hot pot and sear for 3–4 minutes on each side, until a golden-brown crust forms. Remove the chicken to a plate and set aside.
Build the Aromatic Base: Reduce the heat to medium and add the remaining tablespoon of olive oil to the same pot. Add the chopped onion, celery, and carrot. Sauté for 10–12 minutes, stirring occasionally, until the vegetables are completely softened and sweet. Add the minced garlic and fresh thyme, cooking for another 60 seconds until fragrant.
Simmer the Soup: Pour the 8 cups of chicken broth into the pot, using a wooden spoon to scrape up any browned bits from the bottom. Return the seared chicken breasts to the pot. Increase the heat to bring the soup to a gentle boil, then immediately reduce the heat to low. Cover and let it simmer for 20 minutes, or until the chicken is cooked through and registers 165°F (74°C) on an instant-read thermometer.
Toast the Cauliflower: While the soup is simmering, heat the separate medium skillet over medium heat. Add the riced cauliflower to the dry skillet and toast, stirring frequently, for 5–7 minutes. The cauliflower should be dry, lightly golden, and have a nutty aroma. This step prevents a watery soup.
Shred and Combine: Carefully remove the cooked chicken from the pot. On a cutting board, use two forks to shred the meat into bite-sized pieces. Stir the shredded chicken and the toasted cauliflower rice into the soup.
Create the Lemon-Egg Emulsion: In the mixing bowl, vigorously whisk the three egg yolks with the fresh lemon juice and the ground turmeric until the mixture is pale yellow, frothy, and smooth.
Temper and Thicken: Turn off the heat under the soup pot. While whisking constantly, slowly ladle about two cups of the hot broth into the egg-lemon mixture, one ladle at a time. This tempering process gently raises the temperature of the eggs without scrambling them. Once combined, pour the warmed egg mixture back into the main pot of soup, stirring continuously as you pour. The soup will instantly become creamy and opaque. Do not return the soup to the heat.
Finish and Serve: Stir in the ½ cup of fresh parsley and season with additional salt and pepper if needed. Serve immediately in warm bowls.
