In 4-quart saucepan, melt 2 tablespoons of the butter over medium-high heat. Add mushrooms; cook 4 to 6 minutes, stirring occasionally, until browned. Add 2 tablespoons of the butter, the onion and carrots; cook 4 to 6 minutes longer or until vegetables are just tender. Add flour; cook and stir about 1 minute or until mixture is coated and flour begins to brown. Beat in chicken broth and milk. Cook and stir 3 to 4 minutes or until thick and bubbly. Add potato cubes; cook 5 minutes. Stir in chicken, asparagus, 2 tablespoons of the parsley, 1 teaspoon of the thyme, and the salt and pepper. Cook 2 to 4 minutes or until potatoes are tender and mixture is hot. Pour into baking dish.