1. Sprinkle beef evenly with 1/4 teaspoon salt and pepper; dredge in flour. Heat a large skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add beef to pan; cook 4 minutes or until browned, turning occasionally. Add 2 1/2 cups water, scraping pan to loosen browned bits; bring to a boil. Cover, reduce heat, and simmer 1 hour and 45 minutes or until fork-tender. Remove beef from pan; cover and keep warm. Remove cooking liquid from pan; reserve cooking liquid.
2. Heat skillet over medium heat. Add remaining 1 tablespoon oil to pan; swirl to coat. Add carrot and onion; cook 4 minutes, stirring occasionally. Add mushrooms and 1/2 teaspoon salt; cook 5 minutes, stirring occasionally. Add garlic; cook 30 seconds, stirring constantly. Add tomato paste, and cook for 30 seconds, stirring frequently. Stir in reserved cooking liquid; bring to a boil. Reduce heat, and simmer for 6 minutes or until slightly thickened.
3. Bring 3 quarts water and remaining 1 tablespoon salt to a boil in a large saucepan. Add noodles; cook 5 minutes or until al dente. Drain; serve noodles with ribs and sauce.