Grate the zest of the lime and squeeze 1 tablespoon of the juice into a small bowl. Add the coconut milk, soy sauce, ginger, sugar, cornflour and if you are using it, the curry paste then mix together. Set aside.
Heat 1 tablespoon of the oil in a wok or large frying pan over a high heat. Add the onion and garlic and stir fry for 2 minutes until slightly softened. Add the broccoli florets and stalks then the mushrooms and continue to stir fry for a further 2 minutes. Transfer to a bowl.
Heat the remaining oil in the wok. Add the chicken and stir fry for 2 minutes until light golden. Add the coconut milk mixture and return the vegetables to the wok then stir fry for 1 minute to thicken. Cover and simmer for 2 minutes, or until the chicken is cooked through and the vegetables are just tender. Sprinkle with the basil, toss together briefly and serve at once.