In a large pot, cook onion, celery, and rice with a little butter for aboput 5 minute or until vegetables are tender and rice in fragrant. Add half of the broth and bring to a boil over high heat. Reduce heat to medium-high and continue boiling, uncovered for 45 minutes or until rice is tender. As it cooks, add remaining broth as needed (rice will absorb liquid). Sometimes more than 8 cups is needed.
While rice is cooking, mix flour with peper, salt, and herbs. Toss chicken pieces in flour mixture and then cook with some oil in a frying pan until lightly browned. Set cooked chicken pieces aside and reserve flour mixture.
Measure remaining flour mixture and measure out an equal amount of butter. Melt butter in frying pan then slowly whisk in the flour mixture over medium-low to create a smooth roux.
Stir roux into the pot of rice. Then, stir in chicken and milk. Season with salt and pepper.