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For the sauce, heat the oil in a large pan over a medium heat and fry the garlic for 1 min until golden. Add the sugar and vinegar, simmer for 30 seconds, then add the tomatoes. Season with salt and return to a simmer. Bubble for 10 mins, then remove from the heat and set aside. Can be prepared up to two days ahead and chilled.
Heat a griddle or large frying pan over a medium-high heat. Brush the aubergine slices on both sides with the olive oil and season with a little salt, then griddle in batches until the slices are softened and slightly charred (ensure the heat isn’t too high or the aubergine will char before it softens). Transfer the cooked slices to a plate as you go.
To assemble, lay a third of the aubergine slices over the base of a rectangular or square gratin dish, then pour over a third of the sauce. Scatter with a small handful of both cheeses (you’ll want to keep most of the cheese for the top) and half the basil, then top with half the lasagne sheets. Repeat once more, then finish with a final layer of aubergines topped with the rest of the sauce. Scatter over the remaining cheese. After leaving to cool completely, the dish can be covered and chilled for up two days or frozen for up to three months. Defrost fully before cooking.
Heat the oven to 200C/180C fan/gas 6. Put the dish on a baking tray and bake for 30 mins until the cheese is golden and the sauce is bubbling at the edges. Leave to cool for at least 10 mins, then scatter with the reserved basil and serve. The cooked lasagne will keep frozen for up to three months, covered with foil. Defrost thoroughly, uncover and bake as stated above. Or to cook from frozen, bake at 180C/160C fan/gas 4 for 40 mins. Remove the foil, turn the oven up to 200C/180C fan/gas 6 and bake for a further 30 mins.