Heat olive oil in a large soup pot or Dutch oven over medium heat. Add chopped carrots, celery, and shallot or onion. Season with salt and pepper, and sauté for 8–10 minutes until tender. Add garlic, poultry seasoning, and thyme; cook for 1–2 more minutes. Pour in chicken broth, raise heat to high, cover, and bring to a boil. Add halved chicken breasts, reduce heat to low, and simmer for 10–12 minutes or until cooked through. Remove chicken, let cool, then chop or shred. Set aside. Stir rice into the broth. Cover and simmer for 20–30 minutes, stirring occasionally, until the rice is al dente. While rice cooks, melt butter in a small saucepan over medium heat. Whisk in gluten-free flour and cook for 1 minute. Slowly whisk in milk to form a smooth sauce. Stir until thickened, about 1–2 minutes. Slowly pour cream mixture into the soup while whisking constantly. Add shredded chicken and cook for 5 more minutes until thick and creamy. Taste and adjust seasoning. Let sit for 5–10 minutes before serving.