Combine boiling water and mushrooms in a bowl; cover and let stand 30 minutes. Drain, reserving 1/4 cup mushroom liquid, and let cool completely. Discard stems, and finely chop caps; set aside.
Cook pork, gingerroot, and garlic in a large nonstick skillet over medium heat until browned, stirring to crumble. Drain well, and return pork mixture to skillet.
Combine reserved mushroom liquid and cornstarch; stir well. Add to skillet; cook over medium heat 30 seconds or until thickened, stirring constantly. Remove from heat; stir in chopped mushroom caps, cilantro, peanuts, green onions, soy sauce, and sugar.
Working with 1 gyoza skin at a time (cover remaining skins to keep them from drying out), spoon about 2 teaspoons pork mixture into center of each skin. Moisten edges of skin with water. Fold in half, pinching edges together to seal. Place dumplings, seam sides up, on a baking sheet; cover loosely with a towel to keep from drying out.
Line a bamboo steamer with steamed green cabbage leaves. Place dumplings 1/2 inch apart in steamer, and cover with steamer lid. Add water to a large skillet to a depth of 1 inch, and bring to a boil. Place steamer in skillet, and steam dumplings 10 minutes. Remove dumplings from steamer.