Bring a large pot of water to a boil. Add the soba noodles and edamame and cook as the noodle label directs. Reserve 1 cup cooking water, then drain the noodles and edamame and run under cold water until cool.
Whisk the peanut butter, vinegar, soy sauce, ginger, sesame oil and chili sauce in a large bowl. Add 1/2 cup of the reserved cooking water and whisk until smooth. Add the noodles and edamame, season with salt and toss to combine (add more cooking water to loosen, if needed). Divide among shallow bowls and top with the chicken, cucumber and scallions.