Heat a wok over high heat. When the wok begins to smoke, add 1 tablespoon of the peanut oil. Add the carrots and green beans and stir-fry for 1 minute. Splash in 2 tablespoons stock to facilitate cooking and stir-fry for another minute. Transfer to a plate and set aside.
Wipe the wok dry and return to the heat. When the wok is hot, add the remaining 2 tablespoons peanut oil. Add the garlic, ginger and half the chiles and stir-fry for less than 1 minute. Add the curry paste and stir-fry for 1 to 2 minutes. Add the kaffir lime leaves and chicken and stir-fry until the chicken has browned, 2 to 3 minutes. Add the coconut milk and stir-fry for 1 minute. Stir in the remaining 1/4 cup stock, the shiitake mushrooms, scallion light parts, bean sprouts and fish sauce and cook until the chicken has cooked through, 1 to 2 minutes more. Add the green beans and carrots back in and season with salt and white pepper. Transfer to a serving plate and garnish with the remaining chilies, some bean sprouts, the scallion greens and cilantro.