Make the filling: Peel the ginger by scraping it with a spoon, then grate 1 tablespoon. Stem and finely chop the mushrooms. Combine the ginger, mushrooms, scallion, pork, soy sauce, rice wine, sesame oil and cornstarch in a large bowl. Season with salt and pepper. Mix well with your hands until all of the ingredients are incorporated.
Form the dumplings: Place a damp paper towel over the wonton wrappers to keep them from drying out. Remove 1 wrapper and brush with some of the beaten egg.
Make a circle with your thumb and index finger; lay the wrapper on top, nudging it down to create a cup. Add 2 teaspoons filling, then pat the filling down with the back of a spoon.
Fold the overhanging wrapper edges down, leaving the filling exposed. Press the wrapper firmly around the filling. Pat the top and bottom of the dumpling to make it flat.
Top the dumpling with a dried goji berry or 1 each frozen pea and carrot. Repeat to form the remaining dumplings.
Steam the dumplings: Cut out a round of parchment paper to fit in a bamboo steamer and punch holes in the paper to let steam through. Line the steamer with the parchment and brush with vegetable oil; arrange the dumplings in the steamer and cover. Put the steamer in a wok or skillet with a few inches of boiling water, making sure the water does not touch the bottom of the steamer. Steam the dumplings 8 to 10 minutes, or until the pork is cooked through.
Make the sauce: Combine the chile paste and soy sauce in a small shallow bowl. Serve with the dumplings for dipping.