Preheat the oven to 200°C/400ºF/gas 6. Steam the whole aubergines over a pan of simmering water for 30 minutes, then scoop out the flesh and cut it up roughly. Slowly fry the aubergine, garlic, thyme and chilli in the olive oil for around 10 minutes.
Add the tins of tomatoes, chopping them up roughly with a wooden spoon, then add the balsamic vinegar and most of the basil leaves. Bring to the boil and simmer for around 10 minutes until the sauce has reduced and thickened.
Spread a layer of aubergine sauce in a large, shallow dish. Sprinkle over some Cheddar and a handful of Parmesan, then spread over a layer of lasagne sheets. Repeat once or twice more, until your dish is full. Finish with a final sprinkling of Parmesan, a scattering of basil leaves and a drizzle of olive oil.
Place in the oven for 25 to 30 minutes until bubbling and golden.