In a large pot, bring dashi just to a boil, then add the salt, soy sauces, sugar and mirin. Stir to dissolve, then bring the broth back to a gentle boil, and add the chicken pieces. Simmer until the chicken is tender, about 10 minutes, skimming any foam. Add the onions, mushrooms, and cabbage and simmer another 2 to 3 minutes.

Meanwhile, bring a large pot of unsalted water to a rapid boil. Add the noodles, bring back to a boil, and cook, stirring occasionally, until they just begin to soften, about 3 minutes. Drain noodles in a colander and rinse under cold running water while rubbing noodles to remove surface starch. Serve immediately, placing noodles in individual warm, deep bowls. Ladle hot broth over the noodles, and arrange pieces of chicken, onions, and mushrooms on the noodles. Serve immediately.