Heat the olive oil in a large pot over medium heat. Meanwhile, dice the onion (and mince the garlic if using fresh rather than pre-minced). Add the onion and garlic to the hot oil and saute until tender (5 min).
If using fresh carrots and celery, clean and slice them while the onions are sauteing. Add them to the pot and cook for about 5 minutes more.
Add the chicken breast (whole), bay leaf, oregano, thyme, cracked pepper, dry rice, and 6 cups of water to the pot. Do not add salt yet. Bring the pot up to a boil over medium high heat, then reduce the heat to low and let simmer for one hour.
Near the end of the hour, begin to make the creamy milk mixture. In a small pot, melt 2 Tbsp of butter over medium heat. When it is fully melted, whisk in 2 Tbsp of all-purpose flour. Cook and stir the butter and flour mixture for two minutes. Whisk in the milk and allow it to come up to a simmer, whisking occasionally to keep it from scorching on the bottom. Once it reaches a simmer, it will have thickened slightly.
Carefully remove the chicken pieces from the soup and shred or dice with a knife and fork. Rinse and chop the parsley. Add the parsley and the thickened milk mixture to the soup. Season with salt to taste. I used about 2 tsp of salt, but start with a little and add more as needed. I also added a touch more pepper at this point. Serve hot!