Heat oil in a stockpot over medium heat. Add garlic and scallion whites; cook about 3 minutes. Add mushrooms; stir to coat. Stirring occasionally, cook mushrooms until tender, about 3 minutes. Add sake; scrape up any browned bits on bottom of pan. Stir in grated ginger and any accumulated ginger juice. Add 8 cups water; bring to a boil.
Reduce to a simmer; stir in chestnuts, snap peas, and cabbage. Add noodles; stir, and cook until al dente, about 8 minutes.
Stir in scallops, shrimp, soy sauce, salt, and pepper. Cook until seafood is cooked through, about 3 minutes. Serve garnished with scallion greens.