First, make the shredded chicken. (As noted above, you can do this the day before and just take the shredded chicken out of the refrigerator before you begin the final preparations.) Boil 4 qts. of water with one tablespoon of salt. When the water reaches a boil, add the chicken and cook for about 20-25 minutes.
Pour out the water and allow the chicken to cool on a cutting board or in a bowl.
If you are unable to find toasted sesame seeds, toast the sesame seeds in a medium skillet over medium heat, stirring frequently, until golden and fragrant, about 10 minutes. Reserve 1 tablespoon sesame seeds in a small bowl.
In a blender or small food processor, puree the remaining 3 tablespoons sesame seeds, peanut butter, garlic, ginger, soy sauce, vinegar, hot sauce, and sugar until smooth, about 30 seconds. With the machine running, add hot water 1 tablespoon at a time until the sauce has the consistency of heavy cream, about 5 tablespoons; set the mixture aside (it can be left in the blender jar or food processor workbowl). If you wish, add the crushed red pepper for a little zing. (Start with 1/8 of a teaspoon and go from there.).
Bring 5 quarts water and the other tablespoon of salt to a boil in a stockpot over high heat. While you wait for the water to boil, you can shred your chicken.
Using 2 forks, shred the chicken into small, bite-sized pieces and set aside.
Add the noodles to the boiling water and cook them according to the directions on the package (about 4 minutes for fresh and 10 minutes for dried pasta).
Drain the noodles, return them to the pot, then rinse them again with cold running tap water until they are cool to the touch; drain them again.
In a large bowl, toss the noodles with the sesame oil until evenly coated. Add the shredded chicken, scallions, carrot, and sauce; toss to combine. Divide among individual bowls, sprinkle each bowl with a portion of reserved sesame seeds, and cucumber decoration and serve.