Combine chicken, water, and broth in a large pot over medium heat. Cook until chicken is no longer pink in the middle, about 30 minutes.

Meanwhile, heat oil in a skillet over medium heat. Cook carrots, celery, and onion until soft, 5 to 7 minutes. Set aside.

Remove chicken from broth, cut into cubes, and return to broth. Add rice to the broth, reserving seasoning packet, and simmer 20 minutes.

Meanwhile, melt butter in a clean skillet over medium heat. Add flour and reserved rice seasoning packet; cook and stir to create a paste, about 2 minutes. Add milk and whisk to combine. Add milk mixture to the pot with the soup. Stir and simmer until soup thickens, about 5 minutes more.