In a bowl, whisk together chicken broth, peanut butter, 3 tablespoons soy sauce, rice vinegar, sesame oil, and ginger.
In another bowl, combine cornstarch with remaining 2 tablespoons soy sauce, then toss with chicken pieces.
In a large nonstick skillet, heat 2 teaspoons vegetable oil over high heat. Add half of the chicken pieces and cook, stirring, until chicken is cooked through, about 3 minutes. Remove from pan. Repeat with remaining oil and chicken.
In a serving bowl, toss noodles with sauce, cucumber, edamame, and cooked chicken.