Bring 6 quarts water to a boil in a big pot; add in noodles and 1 tablespoon salt; cook until the noodles are just tender, about 3-4 minutes.
Drain thoroughly and toss with the sesame oil; refrigerate until ready to use, at least 2 hours and up to 1 day.
Adjust oven rack to the middle of oven; preheat oven to 400°.
Place chicken breasts on a small, foil-lined jelly-roll pan; brush with vegetable oil and sprinkle generously with salt.
Roast about 35-40 minutes or until meat thermometer reads 160°; cool to room temperature, remove skin, and shred meat into bite-size pieces.
Add ginger, peanut butter, soy sauce, sugar, vinegar, and hot sauce in the bowl of a food processor; process until smooth.
With the motor running, add the water, 1 tablespoon at a time, until the sauce is the consistency of heavy cream.
Scrape the sauce into a bowl; can be covered and set aside for several hours.
When ready to serve, toss the noodles with the chicken, cucumber, peanut sauce, and scallions.
Sprinkle with sesame seeds and serve.