Place chicken in a large Dutch oven. Add water and cook until done. Remove to a plate to cool when done. Reserve water that the chicken cooked in.
Add celery, carrots, onions, salt, white pepper, peas, mushrooms and thyme to reserved chicken broth. Cook over medium-high heat until the vegetables are tender.
Remove thyme stems and add half and half. Lower the temperature to a simmer.
Pick chicken from the bones and add to the vegetable mixture.
In a separate bowl, combine flour, baking powder, and salt. Cut in the butter until mixture is the size of peas. Stir in milk and parsley. Drop dough into simmering stew.
Cook, uncovered, for 10 minutes. Cover and cook 8 to 10 minutes or until dumplings are tender.
