Step 1

Toast sesame seeds in medium skillet over medium heat, stirring frequently, until golden and fragrant, about 10 minutes. Reserve 1 tablespoon sesame seeds in small bowl. In blender or food processor, puree remaining 3 tablespoons sesame seeds, peanut butter, garlic, ginger, soy sauce, vinegar, hot sauce, and sugar until smooth, about 30 seconds. With machine running, add hot water 1 tablespoon at time until sauce has consistency of heavy cream, about 5 tablespoons; set blender jar or workbowl aside.

Step 2

Bring 6 quarts water to boil in stockpot over high heat. Meanwhile, adjust oven rack to 6 inches from broiler element; heat broiler. Spray broiler pan top with vegetable cooking spray; place chicken breasts on top and broil chicken until lightly browned, 4 to 8 minutes. Using tongs, flip chicken over and continue to broil until thickest part is no longer pink when cut into and registers about 160 degrees on instant-read thermometer, 6 to 8 minutes. Transfer to cutting board and let rest 5 minutes. Using 2 forks, shred chicken into bite-size pieces and set aside. Add salt and noodles to boiling water; boil noodles until tender, about 4 minutes for fresh and 10 minutes for dried. Drain, then rinse with cold running tap water until cool to touch; drain again. In large bowl, toss noodles with sesame oil until evenly coated. Add shredded chicken, scallions, carrot, and sauce; toss to combine. Divide among individual bowls, sprinkle each bowl with portion of reserved sesame seeds, and serve.