First, make chicken stock: Cover chicken with water in a large pot or dutch oven. Add the celery, onion, carrots, pepper and thyme. Bring to a boil, then simmer covered, for 45 minutes. Remove chicken and set aside to cool. Strain stock and discard vegetables, set aside. There should be about 2 1/2 cups of stock. Remove meat from chicken and shred.
Heat butter in same pot over medium heat. Cook mushrooms in butter until well done. Add all but 1/4 cup of the strained chicken stock and bring to a simmer. Stir the cornstarch into the 1/4 cup stock and add to the soup, cooking and stirring until it thickens. Slowly stir in the half and half. Return the shredded chicken to the pot, and stir in the cooked rice. Simmer all together 5-10 minutes until flavors blend. Add salt and pepper to taste. May also stir in just a pinch of Herbs du Provence right before serving.