Cook noodles: Cook the soba noodles according to the package instructions, then drain and set aside.
Prepare kale: In a large wok, add the shredded kale with a splash of water and cook for 1-2 minutes until slightly wilted. Cool under running water to retain its vibrant color.
Cook chicken: Heat half the sesame oil in the wok and cook the chicken strips until browned. Remove from the pan and set aside.
Sauté vegetables: Add the remaining oil to the wok. Fry the ginger matchsticks, sliced red pepper, and quartered Brussels sprouts until they start to soften.
Combine and serve: Return the chicken and kale to the wok, add the noodles, and pour in the soy sauce, rice wine vinegar, and lime zest and juice. Stir well, adding a little water if needed to create a sauce that lightly coats the ingredients. Serve hot.