Make the sauce: Combine all the ingredients in a bowl and set aside. Stir again just before using.
Prepare the cups: Heat 2 tablespoons canola oil in a skillet over high heat. Add the chicken and stir-fry until cooked through, about 3 minutes. Add the carrot, celery, bell pepper, peas and red onion and stir-fry about 3 minutes. Add the ginger, garlic, sprouts and shiitakes. Add 1/4 cup of the prepared sauce and cook until thickened, about 30 seconds, scraping up any browned bits. Transfer the stir-fry to a bowl; top with the scallions and peanuts.
Wipe out the skillet. Heat the remaining 4 tablespoons canola oil over high heat. Fry the wontons until golden and crisp, about 15 seconds per side. Transfer to paper towels and season with salt.
Fill the lettuce leaves with the stir-fry. Sprinkle with the sesame seeds. Crumble the wontons on top or serve alongside. Serve with the remaining sauce.