In a large saucepan or Dutch oven, heat olive oil over medium-high heat. Add carrots, celery, onion, salt and pepper, and sauté until slightly softened and onions begin to look transparent, about 5 minutes. Add garlic and cook, stirring, one minute.
Add broth, rice, chicken, bay leaf, dried thyme. Stir and bring to a simmer. Reduce heat to low and simmer for at least 20 minutes or until rice is tender.
Meanwhile, in a small saucepan over medium heat, melt butter. Stir in flour and stir until blended. Slowly stir in milk, whisking to eliminate clumps. Heat and stir until mixture boils and thickens.
When rice is tender, stir white sauce into the soup until blended. Stir in parsley. Check seasoning and add more salt or pepper if needed.
Cover, and keep warm on very low heat until ready to serve.
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Notes
If you prefer, substitute a package of Rice-A-Roni Long Grain & Wild Rice for the half cup uncooked rice. Add the seasoning packet, too, if you want.
Any type of cooked chicken is fine, dark meat or white. Rotisserie chicken is good, too. If you have raw chicken, cut it into small pieces and brown it in 1 tablespoon oil before preparing soup. Remove from pan, set aside and continue with recipe, adding chicken back in with the broth and rice.
Soup is quite thick. If you prefer, add 1 cup water when adding broth.