Toast the sesame seeds in a medium skillet over medium heat, stirring frequently, until golden and fragrant, 7 to 10 minutes. Reserve 1 tablespoon sesame seeds in a small bowl. In a blender or food processor, puree the remaining 3 tablespoons sesame seeds, the peanut butter, garlic, ginger, soy sauce, vinegar, hot pepper sauce, and brown 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).

Bring 6 quarts water to a boil in a stockpot. Add the salt and noodles to the boiling water; boil the noodles until tender, about 4 minutes for fresh and 10 minutes for dried. Drain, then rinse under cold running water until cool to the touch; drain again. In a large bowl, toss the noodles with the sesame oil until evenly coated. Set aside.

Meanwhile, adjust an oven rack to 6 inches from the broiler element; heat the broiler. Spray the broiler pan top with vegetable cooking spray; place the chicken breasts on top and broil lightly browned, 4 to 8 minutes. Using tongs, flip the chicken over and continue to broil until the thickest part is no longer pink when cut and registers about 160 degrees on an instant-read thermometer, 6 to 8 minutes. Transfer to a cutting board and let rest 5 minutes. Using 2 forks, shred the chicken into bite-size pieces. Add the shredded chicken, scallions, carrot, and sauce to the prepared noodles; toss to combine. Divide among individual bowls, sprinkle each bowl with some of the reserved sesame seeds, and serve.