½ pound all-purpose flour, plus more for working the dough (1 3/4 cups)
1¾ cups fine semolina flour, plus more for working the dough
½ teaspoon kosher salt
1¼ cups ice water, plus more as needed
Put the flours and the salt in the bowl of the food processor, and process for a few seconds, to aerate. With the food processor running, pour in the water through the feed tube. Process for about 30 seconds, until a dough forms and gathers on the blade. If the dough does not gather on the blade or process easily, it is too wet or dry. Feel the dough, and add either more flour or more ice water, in small amounts.
Turn the dough out on a lightly floured surface, and knead by hand briefly, until it's smooth, soft, and stretchy. Press it into a disk, wrap well in plastic wrap, and let rest at room temperature for at least 1/2 hour. (Refrigerate the dough for up to a day, or freeze for a month or more. Defrost in the refrigerator, and return to room temperature before rolling.)
Cut the dough in six equal pieces. Keeping it lightly floured, roll each piece through the machine at progressively narrower settings (but not to the narrowest setting), extending it into a strip about ? inch thick, 20 inches long, and 5 inches wide (or as wide as your machine allows). Trim the edges so the rolled strips are even rectangles, and lay them flat on lightly floured trays or baking sheets. Dust the tops with flour, and cover loosely with a kitchen towel. Let the sheets dry for 15 to 30 minutes to make the next steps easier.
To form strangozzi: Lay out one pasta strip on the floured surface in front of you, and roll it up from both short ends, making two fairly tight coils that meet in the middle, like an old-fashioned scroll. With a very sharp knife, slice the scroll crosswise, down through both coils of dough, at 1/4-inch intervals. From a 5-inch scroll, you should be able to cut about twenty cross-sections; each one is a rolled-up strand of strangozzi.
To unfurl the strangozzi, try this clever method that I learned in Umbria: After slicing the scroll, slide the long blade of a serrated knife (or similar thin blade) under the cut pieces, without separating them. Make sure that the knife edge runs exactly under the center line of the scroll, where the two coils meet.
Now lift the knife, and all the cut pieces, off the table. Twist the knife so only the sharp edge, not the flat of the blade, is in contact with the dough. Jiggle the knife gently. If you've centered the blade correctly, the coils of all the cut pieces will begin to unroll, on either side of the knife blade, and soon you'll be holding a score of strangozzi strands. If the dough is sticky in spots, unroll reluctant coils with your fingers. Finally, lower the strands to the work surface, slide them off the knife, gather all into a loose nest, and set it on a floured towel or tray. (If you have problems with this procedure, or don't have a suitable long implement, unfurl the strangozzi by separating the cut pieces by hand and shaking the coils loose one at a time.)
Make strangozzi from the rest of the long dough rectangles. Leave the nests of pasta uncovered, to air-dry at room temperature, until you're ready to cook them (or freeze the nests on a tray until solid, and pack in airtight ziplock bags).