In a medium bowl, mix together the flour, water, and salt until a dough forms. Knead the dough on a floured surface for 5-10 minutes, or until smooth. Cover the dough with a damp cloth and let it rest for 30 minutes.
In another medium bowl, mix together the cabbage, carrots, mushrooms, green onions, garlic, ginger, soy sauce, sesame oil, salt, and pepper to make the dumpling filling.
Roll out the dough on a floured surface until it is about 1/8 inch thick. Use a round cookie cutter or the rim of a glass to cut out circles of dough.
Place a small spoonful of the dumpling filling in the center of each dough circle. Wet the edges of the dough with water, fold the dough in half to form a half-moon shape, and press the edges together to seal. Repeat with the remaining dough circles and filling.
In a large pot of boiling water, carefully add the dumplings and boil for 3-5 minutes, or until the dumplings float to the surface and the dough is cooked through.
Remove the dumplings with a slotted spoon and serve hot, garnished with chopped green onions.