Combine shrimp, mushrooms, soy sauce, Shao Hsing cooking wine, ground pepper, and sesame oil in a bowl. Mix well, cover, and allow it to marinade in the refrigerator for 30 minutes.
Place wheat and tapioca starches in a large bowl.
Bring ¾ cup (180ml) water to a boil. Immediately pour over wheat and tapioca starch. Stir with a spatula to form a sticky dough.
Drizzle vegetable oil over dough and knead for 1 to 2 minutes until a smooth dough forms. Cover bowl with shrink wrap.
Wrapping and Cooking the Dumplings
Remove filling from refrigerator. Mix in sliced green onion.
Lightly flour work surface and rolling pin with some wheat starch. Break a small piece of dough from the bowl and roll it out to about 2mm thick. The thinner the dough, the more translucent the dumpling will appear. Cut out as many circles as possible using a 3-inch cookie cutter. Remove the edges and reuse in the next batch.
Place a wrapper in the palm of your hand and a rounded teaspoon of filling in the center of the wrapper. Fold wrapper into a half circle. Pleat around the edge of the wrapper from the center towards the side. Do the same for the other side. Press to seal and shape into a crescent. Place on a plastic lined plate. Repeat until all dough is used up.
Transfer dumplings onto a parchment lined plate. Fill a deep pot with an inch of water. Place a rack in the center. Bring water to a boil. Place plate with dumplings onto the rack. Cover and steam on medium for 10 minutes.
Cooking the Soup and Putting the Dish Together
While dumplings are being steamed, bring 3 cups (720ml) water in a medium saucepan to boil. Add dashi, soy sauce, mirin, and carrots. Allow it to boil for 2 minutes, then turn off stove.
When dumplings are cooked, remove and place 2 to 3 dumplings in a shallow bowl. Pour some soup over the dumplings. Place a few of the carrots from the soup in the bowl.
Garnish with 2 sprigs of cilantro and serve immediately with ponzu sauce.