In a 5- to 6-quart Dutch oven or pot cook mushrooms in 1 tablespoon oil over medium-high heat until softened. Add onion and an additional 1 tablespoon oil; cook and stir over medium heat until mushrooms and onions are lightly browned. Reduce heat to low. Cook, covered, 10 minutes or until soft and golden, stirring if necessary. Remove from pan.
Sprinkle beef strips with salt and pepper. Heat remaining 2 tablespoons oil in Dutch oven over medium-high heat. Add beef; cook and stir until browned. Remove beef from pan; set aside. Stir in beef broth, wine (if using), Worcestershire sauce, and freshly ground black pepper. Bring to boiling; add noodles and onion mixture. Cook noodles, uncovered, according to package instructions, return meat to Dutch oven the last 3 minutes of cooking.
Meanwhile, if desired, preheat broiler. Arrange toasted bread on a baking sheet and sprinkle with cheese. Broil 3 to 4 inches from the heat for 1 minute or until cheese melts and turns light brown. Top each serving with a toasted bread slice.