Bring a large pot of water to boiling; add noodles. Cook, uncovered, 4 to 5 minutes or until noodles are soft. Drain noodles and rinse with cold water. Snip noodles into short strands.
In the same pot cook mushrooms, white parts of green onions, carrots, ginger, and garlic in hot oil over medium heat 2 to 3 minutes or until mushrooms begin to soften. Add the broth and the 1 cup water; bring to just boiling.
Add spinach, vinegar, and soy sauce. Cook, uncovered, 1 minute more or until spinach is wilted. Stir together miso paste and the 1/3 cup hot water. Stir into soup. Heat through. Top with green onion tops.