Slice the aubergines lengthways into roughly 5 mm slices, lay out on a tray and sprinkle both sides with fine sea salt. Let sit for 30–60 minutes until beads of moisture form, then pat dry with kitchen paper.
Heat 1 tbsp olive oil in a medium saucepan over medium heat. Add the crushed garlic and cook for about 1 minute until fragrant. Stir in the chopped tomatoes, tomato purée and caster sugar. Bring to a gentle simmer, reduce heat to low and cook for 20–25 minutes, stirring occasionally, until slightly thickened. Stir in most of the torn basil leaves and season with salt and freshly ground black pepper. Set aside.
Heat 2–3 tbsp olive oil in a large frying pan over medium-high heat. In batches, fry the aubergine slices for 2–3 minutes per side until golden brown and tender, adding more oil if needed. Transfer the fried slices to a plate lined with kitchen paper to drain.
Preheat the oven to 180 °C (160 °C fan/Gas Mark 4).
Spread a thin layer of tomato sauce over the base of a 20×30 cm ovenproof dish. Arrange a single layer of fried aubergine slices, scatter some mozzarella pieces and a sprinkle of grated Parmesan. Repeat the layers—sauce, aubergine, mozzarella, Parmesan—finishing with sauce, mozzarella and Parmesan on top.
Bake for 35–40 minutes, until the top is golden brown and bubbling.
Allow the parmigiana to rest for 10–15 minutes before serving.