Make the dough: combine the flour and salt in a large mixing bowl, stir, then add 1 ¼ cups of the water. Use your hands to mix until the dough starts to come together in a few large clumps. Start to firmly press and knead the dough, incorporating the loose flour until there’s none left. If necessary, add a little more water, 1 tablespoon at a time, until you can incorporate all of the flour.
Lightly dust a work surface with flour, add the dough, and knead (folding and firmly pressing with your palm, folding and pressing) until the dough looks and feels fairly smooth, about 5 minutes. Form the dough into a ball, wrap it in plastic wrap, and let it rest at room temperature for 1 hour. (Alternatively, once the dough is mostly smooth, you can put it in a zip top bag, seal it, wrap it in a towel, and then knead it more with your feet–I found this to really help create a smoother and more pliable dough, and it’s also the way they do it in Japan!)
On a lightly floured surface with ample room, knead it again for about 2 minutes, lightly dust both sides with flour, then use the rolling pin to roll the dough, occasionally rotating the dough 90 degrees and lightly dusting with flour if it threatens to stick to the pin, into a rough, approximately 17-inch circle with an even thickness (slightly less than ¼-inch). If you are having difficulty rolling, allow the dough to rest for 5 to 10 minutes as needed. This allows the glutens to relax and make it easier to roll out out.
Fold the dough into thirds, then slice widthwise into approximately ⅛-inch-thick noodles. Gently separate the noodles and toss them with a little bit of flour, just so they don’t stick together. Cook right away.
To cook homemade udon: The way you cook homemade noodles is slightly different from the way you cook purchased noodles. Follow these instructions whether you’re planning to serve the noodles hot or cold.
Bring a large pot of water to a boil and prepare a large bowl of icy water. Add the noodles to the boiling water, stirring frequently and adding ¼ cup of fresh water if the water threatens to bubble over, until they’re fully cooked but not mushy, 10 to 12 minutes. (Unlike Italian pasta, they shouldn’t be al dente, but don’t let them get mushy.)
Drain them, then transfer them to the icy water. Briefly and gently rub them with your hands to remove some of the starch. Drain very well.