Add the water and chicken pieces to a large pot over medium-high heat. Bring up to a simmer and cook for approximately 30 minutes.
Add the onion, carrot, celery and garlic to the pot. Continue to cook until the chicken is very tender and the vegetables are soft, approximately another 20 minutes.
Remove the chicken from the pot and set aside until cool enough to handle.
Add the rice and seasoning packets to the pot. Stir together.
In a separate small saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes to cook off the raw flour taste. Slowly whisk in the milk. Bring the mixture up to a simmer and allow it to thicken slightly. Using a ladle, spoon approximately 1 cup of the broth mixture into the milk and stir together. Then pour all of the thickened milk mixture into the large pot, stir, and reduce the heat to medium.
Remove the chicken from the bone and roughly shred.
Add the chicken back into the large pot and stir together. Simmer for 5-10 minutes until the rice is fully cooked and the soup has thickened to a creamy consistency. Season with salt and pepper to taste, and serve.