Trim off and discard mushroom stems. Rinse caps, drain, and cut into 1/4-inch slices.
Cut ham into thin slivers.
Trim off and discard root ends of green onions.
Cut off 4 of the longest green spears from onion tops and set aside. Chop white parts of onions and enough of the remaining tops to make 1 cup. Discard remaining tops.
In a 5- to 6-quart nonstick pan over high heat, combine mushrooms, ham, 1/2 cup chopped onions, water chestnuts, cabbage, and ginger. Stir often until mixture is lightly browned, about 8 minutes. Spoon into a bowl.
Pour 1 cup water and enough of the broth into pan to make liquid 1 inch deep. Bring to a boil over high heat.
Immerse reserved green onion tops in liquid just until wilted, 1/2 to 1 minute. Turn heat to low and set a rack in pan above liquid.
Spoon 1/4 of the mushroom mixture onto the center of each egg roll wrapper. Bring corners of wrapper together and tie with a green onion top to enclose filling.
Gently set bundles side by side on rack. Cover pan, turn heat to high, and steam until bundles are slightly translucent and egg roll wrappers are firm (cut a small piece from top to test), about 5 minutes.
With 2 large spoons, gently transfer 1 or 2 bundles to each of 2 soup bowls.
Add remaining broth and green onions to pan, bring to a boil, then ladle into bowls.
