Wash the aubergines and remove the stalks. Cut them into 1 cm thick slices. Heat the olive oil in a large pan over medium-high heat. Fry the aubergine slices in batches until golden brown. Transfer to kitchen paper, prick several times with a fork, cover with additional kitchen paper, and let rest for 45 minutes.
Peel the onions and garlic. Heat 2 tablespoons of oil in a pan over medium heat, add the onions and garlic, and cook until softened. Add the canned tomatoes with their juice and mash with a fork. Finely chop the parsley and add it along with the thyme, bay leaf, tomato paste, and sugar. Season with salt and pepper. Simmer without a lid for 30 minutes. Remove and discard the bay leaf and thyme sprigs. Puree the sauce if a smoother texture is preferred. Stir in roughly half of the parmesan and adjust seasoning with salt and pepper.
Pat the aubergine slices dry with kitchen paper. Arrange them in a flat baking dish, overlapping like roof tiles. Season with salt and pepper. Pour the tomato sauce over the aubergine slices. Scatter the remaining parmesan on top. Drizzle with 3-4 tablespoons of olive oil. Bake in a preheated oven at 200°C (392°F) or 175°C (347°F) convection for 20 minutes until the cheese is golden.