In a 4- to 5-quart pan over high heat, bring broth to a boil. Add ginger, reduce heat to maintain a simmer, cover, and cook to flavor broth, about 10 minutes. Remove ginger chunks with a slotted spoon or a fine strainer.
Add salmon and simmer until chunks are opaque but still moist-looking in the center (cut to test), about 5 minutes. Add spinach and stir just until wilted.
Increase heat to bring broth back to a boil and slowly stir in eggs. Stir in green onions and add salt and pepper to taste. Ladle soup into bowls immediately.