In a large frying pan; melt butter over medium-high heat. Add beef, onions and mushrooms; cook about 20 minutes, stirring frequently, until beef is browned on all sides. Make sure the onions are carmelized.
Spread noodles in bottom of ungreased 13x9-inch (3-quart) glass baking dish. Spoon beef mixture over noodles.
In the frying pan you used, heat 2 1/2 cups of the broth to boiling over medium-high heat. In small bowl, mix remaining 1/2 cup broth and the flour until smooth. Stir flour mixture into boiling broth. Cook, stirring constantly, until bubbly and thickened. Stir in rosemary, salt and pepper. Pour over beef mixture; stir until well blended. Sprinkle with croutons, then cheese.
.Bake 25 to 35 minutes or until bubbly around edges and cheese is melted.