Preheat the oven to 400F. Slice the two largest aubergines in half lengthways and score them in a 1/8" deep criss-cross pattern, salt and leave to drain upside down over a wire rack for 20 minutes.
Meanwhile, top and tail the other two aubergines, cut into 1" rough cubes, salt and drain in a colander for 20 minutes. While they are draining, cover the tomatoes with boiling water, leave for 30 seconds, drain and peel. Roughly chop the flesh.
Rinse and pat the aubergine dry. Scoop out the flesh of the four halves and place the boat-like skins on a baking tray. Drizzle with a tablespoon of olive oil, season with salt and pepper and roast for 20-25 minutes until soft.
Heat a wide, heavy-bottomed saucepan over a high heat and add all the remaining aubergine flesh with the rest of the oil and some salt. Cook on a high heat for five to 10 minutes until turning golden, stirring every minute or so. Add the red onion, garlic and herbs and cook for a further five minutes, adding a splash more oil if needed. Add the capers and olives, followed by the tomatoes and vinegar. Season with salt, pepper and the sugar, and simmer for about 15 minutes, until thick and tender. Taste and adjust the seasoning.
Spoon a generous helping of caponata into the aubergine boats, sprinkle with the breadcrumbs and parmesan and drizzle with a little extra olive oil. Put back in the oven and roast until the breadcrumbs are golden and crunchy and the parmesan has melted.
Sprinkle with fresh parsley and serve with rice, or just on its own.