Warm 2 tbsp olive oil in a 3-quart / 2.8 L saucepan over medium heat. Add the garlic and pepper flakes and cook for 30 to 60 seconds until fragrant but not browned.

Add the crushed tomatoes and 1/2 tsp salt. Simmer gently for 15 minutes, stirring now and then, until the sauce leaves a trail when you drag a spoon across the bottom. Stir in half the basil, torn, and take the pan off the heat.

While the sauce simmers, brush both sides of the aubergine slices with the 60 ml olive oil and season with 1/2 tsp salt and the pepper. Heat a 36 cm / 14-inch ridged grill pan or outdoor grill over medium-high until very hot.

Grill the slices in 3 batches, about 7 or 8 slices at a time, for 3 to 4 minutes per side. Pull them when they have dark grill marks, are fully soft, and bend without cracking. Pale slices that still squeak will turn leathery in the oven and release water. Start heating the oven to 200 C / 390 F when the last batch goes on.

Spread about 120 ml / 1/2 cup of the sauce over the bottom of a 23 x 33 cm / 9 x 13-inch baking dish. Add a layer of aubergine, overlapping the slices slightly, then top with a third of the remaining sauce, a third of the mozzarella, and a third of the parmesan. Repeat twice more, ending with cheese on top.

Bake uncovered for 25 minutes, until the sauce bubbles at the edges and the mozzarella is melted with golden spots. Rest for 10 minutes, scatter the remaining basil over the top, and cut into 4 squares.