In a large heavy kettle, melt butter or margarine. Add carrots, celery, onion, green pepper, mushrooms, and ham; cook over medium heat until vegetables are crisp tender, about 10 minutes, stirring occasionally. Do not brown.
Stir in flour and cornstarch; cook, stirring constantly, about 3 minutes. Add broth and cook, stirring, until slightly thickened. Add milk, paprika, cayenne, and mustard.
Stir in cheese gradually, stirring until cheese is melted. To avoid curdling, do not allow soup to boil after cheese is added. Season to taste with salt and pepper. Serve piping hot.