Sprinkle chicken with 1/2 teaspoon pepper, 1/4 teaspoon salt, and thyme. Coat a Dutch oven with cooking spray; add oil. Place over medium heat until hot. Add chicken and onions; cook until chicken is browned on all sides. Remove chicken, and set aside.
Add 2 3/4 cups chicken broth and wine to pan; bring to a boil. Cover, reduce heat and simmer 15 minutes. Return chicken to pan; cover and simmer 10 minutes.
Coat a large nonstick skillet with cooking spray; place over medium-high heat until hot. Add mushrooms, and saute until tender. Add garlic, and saute 1 minute.
Combine remaining 1/4 cup broth and flour, stirring with a wire whisk until smooth. Gradually add to chicken mixture, stirring well. Stir in mushroom mixture, sour cream, white pepper, and 1/8 teaspoon salt. Cook over low heat 5 minutes or until thoroughly heated, stirring frequently. Sprinkle with parsley.