Bring water to boil with ginger slices. Add chicken breast and poach until cooked through (internal temperature 165°F/74°C). Remove and let cool, then shred into thin strips.
Cook noodles according to package instructions. Drain and rinse under cold running water until completely cooled.
Blanch bean sprouts in boiling salted water for 1 minute, then drain and cool in ice water.
Prepare the sauce: Mix sesame paste, chili oil, rice vinegar, soy sauce, sesame oil, minced garlic, and Sichuan peppercorn powder in a small bowl. Add sugar and mix well until smooth.
Arrange cooked noodles on individual plates or a serving platter. Top with shredded chicken, cucumber strips, bean sprouts, and shredded carrots in neat rows.
Pour the sauce generously over the noodles and toppings. Sprinkle roasted peanuts and white sesame seeds on top.
Garnish with green onion pieces. Serve immediately while chilled, tossing well before eating to evenly distribute the sauce.
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