Put a large pot of water on to boil for cooking pasta.
Meanwhile, place chicken in a skillet or saucepan and add enough water to cover; bring to a boil. Cover, reduce heat to low and simmer gently until cooked through and no longer pink in the middle, 10 to 12 minutes. Transfer the chicken to a cutting board. When cool enough to handle, shred into bite-size strips.
Whisk peanut butter, soy sauce, garlic, chile-garlic sauce and ginger in a large bowl.
Cook pasta in the boiling water until not quite tender, about 1 minute less than specified in the package directions. Add vegetables and cook until the pasta and vegetables are just tender, 1 minute more. Drain, reserving 1 cup of the cooking liquid. Rinse the pasta and vegetables with cool water to refresh. Stir the reserved cooking liquid into the peanut sauce; add the pasta, vegetables and chicken; toss well to coat. Serve warm or chilled.