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Preheat oven to 200°C fan. Line a large baking tray.
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Arrange the eggplants cut-side down on the tray with the garlic cloves. Drizzle generously with olive oil and season with salt and pepper. Roast for 20–25 minutes.
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Remove the tray, add the cherry tomatoes, drizzle with a little more olive oil and return to the oven for a further 20 minutes, or until the eggplants are soft and the tomatoes are blistered and jammy.
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Scoop the eggplant flesh into a bowl, discarding the skins. Squeeze the roasted garlic from its skins and add to the bowl. Blitz with a good drizzle of olive oil, salt and pepper until smooth and velvety. Adjust seasoning to taste.
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Meanwhile, cook the prosciutto in a dry frying pan over medium heat until golden and crisp. Transfer to paper towel to cool and harden. (Alternatively, bake in the oven at 200°C for 10–12 minutes.)
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Bring a large pot of salted water to the boil. Cook the pasta as per packet directions. Using tongs, transfer the pasta directly into a large bowl or pan with the eggplant purée, allowing some pasta water to come along. Toss well, adding extra pasta water as needed to create a glossy sauce.
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Transfer to a large serving platter. Top with roasted tomatoes, shards of crispy prosciutto and torn basil leaves. Finish with a drizzle of olive oil and generous shavings of Parmigiano Reggiano.
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Serve and enjoy with family and friends!