Bring the stock, garlic, ginger and chilli to the boil over moderate heat in a large frying pan. Reduce to a simmer, add the chicken, cover and cook, turning the chicken over once, for 10 minutes or until the chicken is cooked through. Transfer the chicken to a plate, reserving the cooking liquid. When it is cool enough to handle, shred the chicken.
Meanwhile, cook the noodles in a large saucepan of boiling water according to the packet directions, until firm-tender. Add the beans and carrots for the last minute of cooking time; drain.
Whisk together the brown sugar, soya sauce, oil and the reserved cooking liquid in a large bowl. Add the shredded chicken, noodles, beans, carrots and the cabbage, tossing to combine. Serve at room temperature or chilled.