Pat the chicken dry with paper towels, then season generously with salt and pepper. Heat 1 teaspoon of the oil in a large nonstick skillet over medium-high heat until just smoking. Add the chicken and cook until browned on the first side, about 5 minutes. Flip the chicken over, add ½ cup water, and reduce the heat to medium-low. Cover and continue to cook until the thickest part of the breast is no longer pink and registers about 165 degrees on an instant-read thermometer, about 5 minutes. Transfer the chicken to a plate and cover with plastic wrap. Poke a few vent holes in the plastic wrap and refrigerate the chicken while preparing the other ingredients. When the chicken is cold, shred it into bite-sized pieces.
Meanwhile, shake the vinegar, remaining 1 tablespoon oil, hoisin, soy sauce, and ginger together in a jar with a tight-fitting lid; set aside.
Toss the shredded chicken, carrot, cabbage, red pepper, sprouts, scallions, and cilantro in a large serving bowl. Shake the dressing to recombine, pour it over the chicken salad, and toss to coat. Sprinkle with the chow mein noodles before serving.