In a large bowl, whisk together flours until evenly combined. Pour water over the flours and, using your fingers, toss and rub the flours with the water until crumbly, like muddy sand.
Scrape the dough onto a work surface and press and knead until smooth, about 6 minutes. Press the dough into a disk, then rotate the disk clockwise as you pinch portions of the dough on top of the disk and fold them over in a counterclockwise motion to form pleats.
Arrange the dough pleated-side-down and mold into a cone. Flatten the dough around its perimeter until it is 1⁄2-inch thick, keeping a slight bump in the center of the disk. Transfer the dough to a work surface dusted lightly with tapioca starch and lightly dust the dough with starch.
Using a thin rolling pin or wooden dowel dusted with more starch, roll the dough using back-and-forth strokes, rotating it as needed. (You can also use a pasta machine for this task by cutting the disk into quarters and feeding each through the machine on its thinnest setting.)
Once the dough is flattened to 1⁄18-inch thick (you can roll the dough around the rolling pin to check for even thickness), dust the flattened dough generously with starch and fold in half. Dust the sheet with more starch and fold it again in the same direction to make four layers.
Using a sharp slicing knife, slice the dough into 1⁄16-inch-thick noodles and toss in the starch to ensure they don't stick together. Use the cut soba noodles immediately or transfer to a parchment paper-lined baking sheet, cover with plastic wrap, and refrigerate for up to 3 days.