How to Make It
Combine boiling water and mushrooms in a bowl; cover and let stand 30 minutes. Drain; discard stems, and mince caps.
Combine chicken and next 7 ingredients (chicken through salt) in a large bowl; add 3 tablespoons minced mushrooms (set remaining mushrooms aside), and stir well. Set chicken mixture aside.
Working with 1 gyoza skin at a time (cover remaining skins to keep them from drying out), moisten 4 equally spaced points around edge of skin with water. Spoon about 1 teaspoon chicken mixture into center of each circle. Bring 4 moistened points over filling to center, pressing fingers together. Push fingers toward center of circle to form 4 pockets (if pockets do not open enough for chicken mixture to be visible, use the tip of a knife to separate the dough and open pockets). Working with 1 dumpling at a time, fill first pocket with minced mushrooms, second pocket with peas, third pocket with carrot, and fourth pocket with 1 piece of shrimp. Place filled dumplings on a large baking sheet sprinkled with 2 teaspoons cornstarch; cover loosely with a towel to keep from drying out.
Line a bamboo steamer with steamed cabbage leaves. Place filled dumplings 1/2 inch apart in steamer, and cover with steamer lid. Add water to a large skillet to a depth of 1 inch; bring to a boil. Place steamer in skillet, and steam dumplings 10 minutes; remove dumplings from steamer. Serve dumplings with dipping sauces.