About 1 1/4 hours before serving: discard bones and skin from chicken; cut meat into bite-sized pieces.
Cut potatoes into 1/2-inch chunks; set chicken and potatoes aside.
Preheat oven to 400.
Slice mushrooms, carrots and celery; chop onion.
In a 12-inch skillet over high heat, in hot salad oil, cook mushrooms, carrots, celery and onion for 5 minutes, stirring frequently, until vegetables soften slightly; stir in flour, salt, chicken bouillon, dried thyme and pepper until vegetables are well coated.
Add chicken, potatoes and 3 1/3 cups milk to mixture in skillet; over high heat, heat to boiling.
Boil one minute, then spoon chicken mixture into a deep 3-quart casserole; bake, uncovered, for 10 minutes.
Meanwhile, prepare corn-muffin mix according to label, using egg and remaining 2/3 cups milk.
After chicken mixture has baked 10 minutes, remove casserole from oven and spoon cornmeal mixture over the top of chicken mixture.
Bake 15 minutes longer or until cornmeal crust is golden and chicken mixture is hot and bubbly.