Cook the vegetables. Heat the extra virgin olive oil in a Dutch oven or large soup pot over medium heat. Add the carrots, celery, and shallot or onion, then season with salt and pepper. Sauté for 8–10 minutes, until the vegetables are tender. If you want to speed up the softening process, add a splash of stock or water and cover the pot with a lid. Stir in the garlic, poultry seasoning, and dried thyme and sauté for another 1–2 minutes, until fragrant.
Cook the chicken. Pour the chicken stock or broth into the pot. Increase the heat to high, cover, and bring the liquid to a boil. Add the chicken breast halves, replace the lid, and reduce the heat to low. Simmer for 10–12 minutes, or until the chicken breasts are cooked through. Transfer the chicken to a cutting board. Once cool enough to handle, chop or shred it. Place the chicken in a bowl or storage container and refrigerate until it is time to return it to the soup.
Simmer the rice. Meanwhile, stir the rice into the pot. Cover and simmer until the rice reaches an al dente texture, about 20–30 minutes depending on the brand. Stir occasionally as it cooks.
Prepare the creamy thickener. When the rice has roughly 10 minutes of cooking time remaining, melt the butter in a small saucepan over medium heat. Sprinkle in the gluten free flour and whisk continuously for 1 minute. Slowly drizzle in the milk while whisking so the mixture remains smooth. Switch to a spatula and continue cooking and stirring for 1–2 minutes, until thick and creamy. Remove the saucepan from the heat, cover, and set aside.
Finish the soup. When the rice is al dente, gradually pour the butter, flour, and milk mixture into the soup while whisking constantly. Return the chicken to the pot and cook for 5 more minutes, stirring frequently, until the rice is tender and the soup has thickened. Taste and add more salt and pepper if necessary. Remove from the heat and allow the soup to stand for 5–10 minutes so it can thicken slightly before serving.