Step 1 – In a large bowl, toss together the tomatoes and salt. Let sit until juicy, about 10 minutes.
Step 2 – Meanwhile, bring a large pot of water to a boil. Cook the noodles according to package instructions, drain and rinse under cold water. Set aside.
Step 3 – Add the vinegar, soy sauce, sugar, garlic, mustard and sesame oil to the tomatoes, and toss with a spoon until well combined. Stir the filtered water into the tomatoes and sprinkle the surface of the broth with the sesame seeds, radishes and scallions.
Step 4 – Right before serving, add the ice to the broth. Divide the noodles among bowls, and ladle in the broth and any unmelted ice, making sure each serving gets a nice sprinkling of tomatoes, radishes, scallions and sesame seeds.