Step 1Place the buckwheat and all-purpose flours in a gallon-size resealable plastic bag, along with the cold water. Seal the bag closed and work the water into the flours, massaging the mixture with both hands, until well combined. Continue to work the dough until it forms a single mass. (Press and rub the sides of the bag against the dough to pick up as much as you can of any dough that's sticking to the bag.)
Step 2Remove the dough from the bag to a cutting board. Working quickly and using the heels of your hands, continue to knead firmly until a smooth dough forms. (If the dough feels dry, lightly wet the tips of your fingers with more cold water, brushing them against the surface of the dough and continue kneading until smooth). The final dough will be soft, smooth and not sticky. This will take about 4 to 5 minutes. Form the dough into a smooth ball.
Step 3Cover the dough loosely with plastic wrap while you sprinkle cornstarch on a large cutting board. Place the ball on the board and lightly sprinkle cornstarch over the top. Using your palm and the heel of your hand, flatten the ball into a disk about one-half inch thick.
Step 4Use a rolling pin to roll the disk into a rectangle (about 20 by 12 inches) one-eighteenth-inch thick. Generously sprinkle cornstarch over half of the dough and fold the other half of the dough over, like a book (the cornstarch will keep the dough from sticking together as it is cut). Generously dust another crosswise half of the dough with cornstarch and fold again.
Step 5Starting along the short, folded side of the dough, slice it into very thin (about one-sixteenth-inch) noodles. Keep the noodles loosely covered with plastic wrap while you boil the water for cooking.
Step 6To cook the noodles: Bring a large pot of water (at least 2 gallons) to a boil over high heat. Gently drop the soba into the boiling water. Keep the water boiling vigorously to prevent the noodles from sticking together. Cook the noodles to al dente, about 90 seconds (timing will vary depending on the thickness of the noodles).
Step 7Immediately remove the noodles to a strainer set in a bowl of ice water to stop the cooking. Prepare a second bowl of ice water and transfer to the second bowl to remove any surface starch and cool completely. Drain the noodles. Serve with soba-tsuyu dipping sauce or walnut dipping sauce on the side, along with condiments.