Steam the whole aubergines in a colander over a pan of simmering water for 30 minutes, then halve, scoop out and roughly chop the flesh.
Meanwhile, peel and finely slice the garlic and pick the thyme leaves.
Heat the olive oil in a large frying pan over a medium heat, add the garlic, thyme and aubergine, then crumble over the chilli and cook for 10 minutes.
Tip in the tomatoes, breaking them up with a spoon, then add the balsamic vinegar.
Pick in most of the basil leaves, then bring to the boil and simmer for 10 minutes, or until the sauce has thickened and reduced, then taste and season to perfection with sea salt and black pepper.
Grate the Parmesan and half the Cheddar, roughly slicing the remaining Cheddar.
To layer up, spread a layer of tomato sauce across the base of an ovenproof dish. Sprinkle lightly with grated cheese, cover with a layer of lasagne sheets, then repeat the layers twice more. Finish with a layer of sauce, sprinkle with the remaining grated cheese and dot over the slices of Cheddar.
Bake at the bottom of the oven for 25 to 30 minutes, or until golden and bubbling.
Pick over the remaining basil, drizzle lightly with extra virgin, then serve.