Mix broth, carrots, celery, uncooked rice, onions, thyme, and 1/2 teaspoon pepper & ½ tsp salt in pot.
Bring to a boil; reduce heat, cover and simmer 1 hour, until the rice is tender.
When rice is about tender, melt butter in a smaller saucepan, stir in the flour, then the half-half and wisk over medium heat until it thickens (2-3 min).
Slowly add half-half of the mixture to the larger soup pot rice mixture, stirring constantly. Stir in the chicken and heat through.