Combine first 5 ingredients in a Dutch oven; bring to a boil. Add chicken; cover, reduce heat, and simmer 30 minutes or until tender. Remove chicken with a slotted spoon; bring broth mixture to a boil. Cook until reduced to 4 cups (about 30 minutes). Pour reduced broth mixture into a zip-top plastic bag. Snip off 1 corner of bag; drain 2 cups broth mixture into a 2-cup glass measure, stopping before the fat layer reaches the opening. Reserve remaining broth mixture for another use. Discard fat.
Remove chicken from skin and bones; shred with 2 forks to measure 4 1/2 cups meat. Place chicken in a 13 x 9-inch baking dish coated with cooking spray.
Preheat oven to 400°.
Place a large skillet coated with cooking spray over medium-high heat. Add mushrooms; sauté 5 minutes. Remove from skillet. Place flour in skillet; gradually add milk, stirring with a whisk until blended. Add 2 cups broth mixture; cook over medium heat 12 minutes or until slightly thick, stirring constantly with a whisk. Stir in mushrooms, salt, and pepper; pour over chicken.
Combine the stuffing mix and butter in a bowl, and sprinkle over the mushroom mixture. Cover and bake casserole at 400° for 20 minutes. Uncover and cook casserole 10 minutes or until bubbly.