Bring a large pot of salted water to a boil. Add soba noodles and cook 3 minutes. Add edamame and carrots to noodle water and cook 3 minutes more. Drain.
In a large bowl, whisk together peanut butter, salad dressing, and 1/4 cup very hot water. Add drained noodle mixture to peanut sauce and toss to coat well.
Add chicken and cucumber slices and toss again. Serve warm or at room temperature.