Preheat oven to 350°.
Heat oil in a large nonstick skillet over medium heat. Add mushrooms, celery, onion, thyme, and garlic; cook 10 minutes or until tender, stirring occasionally. Stir in turkey, salt, and black pepper. Remove mixture from heat, and cool slightly.
Combine milk, mustard, red pepper, and eggs in a large bowl, stirring with a whisk. Stir in mushroom mixture and bread; toss well. Spoon into an 11 x 7-inch baking dish coated with cooking spray. Sprinkle with cheese. Bake at 350° for 1 hour or until golden brown. Let stand 10 minutes before serving.