In a heavy saucepan, combine the chicken and water, bring to a boil, reduce the heat to low and carefully skim off any scum. Add onion, carrot, celery, bay leaf, peppercorns, and salt. Cover and simmer the chicken for 45 minutes to one hour. Remove the chicken from the broth and set aside to cool. Strain the remaining broth, discarding the solids, and place in the refrigerator to chill, or use fat separator to remove all fat from broth. When the chicken is cool enough to handle, remove the meat from the bone and tear or shred into bite sized pieces. Cover and refrigerate until needed.
Meanwhile, in a large saucepan, heat the olive oil until sizzling, add the onions, celery, carrots and garlic and cook over medium high heat for about 5 minutes. Lower the heat to medium, add the mushrooms and cook for 5 minutes more. Add the flour, salt and pepper and stir to combine, cooking for 2 minutes. Add the chicken stock (remove the fat that has accumulated at the top of the stock) and the evaporated skim milk. In a small bowl, whisk together the sour cream and the cornstarch. Add to the soup, stirring until thickened and heated through. Add the cooked rice, shredded chicken and thyme, stirring until thick and bubbly. Serve at once.