Bring the water to a boil in a medium pot then add the rice and 1 tablespoon of olive oil and stir. Cover and reduce to a simmer. Cook for about 30 minutes until most of the water is absorbed but rice is still slightly undercooked. Drain and set aside.
Melt the butter and the remaining tablespoon of olive oil in a large stock pot or dutch oven over medium heat. Add the onion, celery, and carrots and cook for 4-5 minutes until the vegetables are softened.
Add the flour and cook for 1-2 minutes, stirring constantly, until the flour is lightly browned.
Stir in the chicken broth and milk and continue stirring for another minute. Add the salt, pepper, thyme, chicken, and reserved rice and bring to a simmer. Cook for 10-15 minutes until the soup is slightly thickened and the rice is fully cooked.