Combine water, mushrooms, 1/2 the green onions, carrot, onion, soy sauce, garlic, Sriracha sauce, sesame oil, rice wine vinegar, white pepper, black pepper, and flavor packets from ramen in a large pot. Bring to a boil. Add noodles and cook, occasionally stirring, until noodles are tender yet firm to the bite, about 5 minutes.

Reduce heat to a simmer and cook until vegetables are tender, about 10 minutes. Serve topped with remaining green onions.