In a microwave-safe bowl, combine the onion, garlic, oil, tomato paste and thyme; cook for 3-4 minutes, stirring every minute, until onions are tender. Pour into slow cooker.
Sprinkle the chicken with salt and pepper. Stir the chicken, mushrooms, broth, vinegar and tapioca into the slow cooker.
Cover and cook on LOW for 4-6 hours.
Remove chicken; use two forks and break it into large pieces. (this is optional, I left mine whole)
Remove fat from surface of sauce with a large spoon.
Place chicken back into the slow cooker. Add the evaporated milk, cheese and parsley.
Salt and pepper to taste and serve over rice or pasta.