Preheat oven to 425 degrees F. For corn bread, prepare muffin mix according to package directions (do not bake); set aside.
In a medium saucepan, cook carrot, onion, sweet pepper, and celery in hot oil over medium heat until tender. Stir in soup, milk, and mustard; stir in sausage. Cook and stir until bubbly. Transfer mixture to an ungreased 2-quart rectangular baking dish. Pour corn bread batter over hot sausage mixture, spreading evenly.
Bake, uncovered, for 20 to 25 minutes or until a toothpick inserted in corn bread comes out clean. If desired, sprinkle each serving with cheese. Makes 6 servings.