Bring a large pot of salted water to a boil and cook chickpea noodles according to package instructions. Drain and rinse immediately with cold water to stop the cooking process.
Place chicken breasts in a skillet, cover with water, and simmer over medium heat for 12-15 minutes until the internal temperature reaches 165°F (74°C). Remove and shred using two forks.
In a medium mixing bowl, whisk together the peanut butter, tahini, soy sauce, rice vinegar, honey, ginger, garlic, and sesame oil.
Slowly whisk warm water into the sauce one tablespoon at a time until a smooth, pourable consistency is achieved.
In a large bowl, combine the chilled chickpea noodles, shredded chicken, steamed edamame, and julienned cucumber.
Pour the sesame dressing over the ingredients and toss thoroughly with tongs until evenly coated.
Garnish the dish with sliced green onions and toasted sesame seeds.
Serve immediately or refrigerate for 30 minutes to allow flavors to meld.