Bring a medium pot of water to a boil. Add soba and cook, according to package directions, until al dente. Drain. Divide among four bowls.
Combine dashi or broth, soy sauce, sugar and mirin in a large saucepan over medium heat. Bring just to a boil. Add chicken; cook 2 minutes. Add spinach, by handfuls. When spinach is just wilted, ladle soup over soba.