In a food processor, finely chop the ginger. Transfer to a saucepan, add the salt and water and bring to a boil. Simmer over moderate heat for 10 minutes. Strain the ginger broth through a fine sieve, discarding the ginger. Return the broth to the saucepan and bring to a boil. Add the shrimp and cook over high heat just until opaque, about 1 minute. Remove the shrimp to a plate with a slotted spoon and let cool. Let the cooking liquid cool too, then add the shrimp and refrigerate until chilled, at least 30 minutes. Drain the shrimp and pat dry.
In a large pot of boiling water, cook the cabbage until tender, about 2 minutes. Drain and let cool under running water, then dry the cabbage leaves thoroughly with paper towels.
Cut the cabbage into 6-by-1-inch strips. Set a shrimp at one end of a cabbage strip, top with a scant 1/8 teaspoon of the black bean-chili sauce and a few of the sprouts and roll up the shrimp in the cabbage strip. Repeat with the remaining ingredients. Serve chilled or at room temperature.
The recipe can be prepared through Step 2 up to 6 hours ahead. Refrigerate the shrimp; leave the black bean sauce and cabbage at room temperature.