Prepare Chicken - Boil enough water in a wok to cover chicken. Add chopped scallions, crushed ginger and garlic, and whole peppercorns. Add chicken and simmer until cooked through, about 15 minutes or reaches 150°F (66°C) on an instant read thermometer. When cool enough to handle, remove skin and bones from chicken and shred into thin, long pieces by hand.
Prepare Dressing - In a sealed container or jar, combine minced garlic, ginger, ground sichuan pepper, sesame paste, red oil, vinegar, soy sauce, sugar, and sesame oil. Cover container and shake to combine.
Serve - If using bean sprouts, heat 1 tbsp toasted sesame oil in a wok. Stir fry bean sprouts for 3 minutes. If using lettuce, shred first and mix with 1 tbsp sesame oil. Place sprouts or shredded lettuce onto the middle of a serving platter. Fan chicken shreds over sprouts. Rim dish with cucumber sticks. Pour shaken dressing over chicken shreds and garnish with sesame seeds. Serve.