Begin by making the wonton dough: In a large bowl, stir together the flour and salt. Add the eggs and stir until a dough forms. Add one to two tablespoons of water if the dough is too dry. Dump the mixture on to a floured board and knead the dough for 10 minutes.
Cover the dough loosely with a floured cloth and let it rest for thirty minutes. Meanwhile, make the filling by combining in a food processor: the cabbage, 1 tsp. of ginger, the garlic, pork, wine, 1 tsp. of the soy sauce, the scallions, and the sesame oil. Pulse until everything is minced together.
Divide the dough in half, and roll each half out as thinly as possible. Use a three inch sharp cookie cutter to cut about 30 circles. On each dumpling, place a teaspoon of filling. Wet the edges with water, and fold the circle in half. Bring the two points together to form a wonton and seal with a dab of water. Repeat with rest of dumplings, placing them on a floured tray.
Bring a large pot of salted water to a boil, and cook the dumplings for about 12 minutes on a low boil.
Meanwhile, in a Dutch oven, heat the butter and cook the mushrooms until they're brown. Then add the rest of the soup ingredients and simmer for 10 minutes.
Drain the wontons and serve with the soup. Top each bowl with the garnishes.