In a bowl, gently mix the ground chicken and burger ingredients (except buns and peanuts) just until combined. Form into 6 patties slightly wider than the buns and refrigerate while you prepare the sauce.
Heat sesame oil in a saucepan over medium heat. Add the dried chiles and ground Sichuan pepper and cook until fragrant. Stir in the ginger, garlic, and scallions, then add the soy sauce, Shaoxing wine (if using), brown sugar, black vinegar, and chicken stock. Simmer until slightly reduced, then whisk in the corn starch slurry, and cook until thick and glossy. Keep warm and set aside.
Preheat the grill or pan to medium-high heat. While the grill heats, toss the slaw ingredients in a bowl and set aside.
Lightly oil the patties and cook until cooked through and lightly charred. Transfer to a plate and let rest briefly.
Spread a light swipe of kung pao sauce on the toasted bottom bun. Then, add the chicken patty, drizzle with more sauce, then top with slaw and crushed peanuts. Finish with the top bun and serve immediately.