Preheat the oven to 190°C. Heat the oil in a flameproof casserole dish or deep frying pan, add the onion and garlic and cook, stirring, for 2 minutes over a medium heat. Add the mushrooms and cook for 3–4 minutes until lightly coloured, stirring frequently.
Stir in the eggplant, cook for 2 minutes, then add the tomatoes and stir over a medium-high heat for 2 minutes to combine. Reduce the heat and leave to simmer for 15 minutes until thick. Stir in the basil and season to taste.
If using a flameproof casserole dish, lay the chicken in the sauce. Alternatively, arrange the chicken in a single layer in a deep ovenproof baking dish or casserole dish and spoon the sauce over the top. Cover and bake for about 25 minutes or until the chicken is thoroughly cooked.
Towards the end of the cooking time, preheat the grill. Uncover the chicken and sprinkle the cheese over the top of the dish. Put the dish under the grill for 1–2 minutes until the cheese is melted and bubbling.