1. MAKE THE DOUGH: Place 2 cups of the flour in a large bowl and keep the remaining ¼ cup handy. Add the oil and 1 cup of water and mix until all the liquid has been absorbed. Work the dough with your hands into one manageable ball. If the dough is wet and sticky, work in some of the remaining flour, a little at a time, until it no longer sticks to your finger.
2. Place the dough on a floured surface and knead for 2 to 3 minutes. The dough will be somewhat firm and elastic. Coat the dough with a little oil, return it to the bowl, cover with a clean kitchen towel, and let it rest for 30 minutes.
3. MAKE THE FILLING: While the dough is resting, drain the mushrooms and squeeze out any excess liquid. Remove and discard the stems, finely chop the mushroom caps, and place them in a medium bowl along with the other filling ingredients. Mix together well and keep refrigerated until you are ready to assemble the dumplings.
4. MAKE THE SOY-VINEGAR SAUCE:
Combine the soy sauce, rice vinegar, sugar and sesame oil in a small bowl. Stir until the sugar has dissolved. Pour into small dipping bowls and add the scallion just before using. This sauce will keep for up to 1 week in the refrigerator in a tightly sealed container.
5. ASSEMBLE THE DUMPLINGS: Line a tray with a kitchen towel and sprinkle with a little flour. Also have ready the dough and filling.
6. Knead the dough once or twice on a floured surface, divide it into 4 equal pieces, and set 3 of them aside under a kitchen towel. Shape the remaining piece into a ball, then roll it out until about 1/8 inch thick. Sprinkle with flour if it gets sticky. Let the rolled-out dough relax for a couple of minutes before cutting out rounds.
7. Using the cookie cutter, cut out as many rounds as you can, usually 7 or 8. Collect the leftover dough and put it aside under a damp kitchen towel for later use.
8. Pick up 1 dough round and pleat and pinch one side. This will pull the round into a bowl shape. Place a rounded tablespoon of filling in the center. Bring together the edges, pushing out any air, and pinch along the top of the dumpling to seal. Place the dumpling, pleated side up, on a prepared tray. Repeat with the remaining rounds and filling. Keep the assembled dumplings in a single layer on the tray, covered with a kitchen towel as you work. Once you have assembled the first batch of dumplings, continue with the remaining dough and filling. Combine all the dough scraps, knead them into a ball, roll it out, cut out as many rounds as you can, and fill and fold those too.
9. Set aside the number of dumplings that you would like to cook and keep the rest frozen for up to 6 months (see Notes).
10. COOK THE DUMPLINGS: Generously coat a large skillet with oil and heat it over medium heat. Arrange the dumplings in the pan, pleated side up, leaving a little space around each. Cook until the bottoms just begin to brown. Choose an empty spot in the pan and pour in enough of the flour and water mixture to reach about a third of the way up the sides of the dumplings. Cover and simmer for 10 minutes. (If cooking frozen jiao zi, add them directly to the oil and increase the cooking time by 3 minutes. Do not allow the dumplings to thaw before cooking.)
11. Remove the cover so that any remaining liquid can cook off completely. Continue cooking until you hear a strong sizzling sound and the bottoms of the dumplings are golden brown and crisp.
12. Remove the jiao zi with a small spatula and place them on a plate. Serve immediately while you cook any remaining jiao zi. Serve with the Soy-Vinegar sauce.