Trim top and bottom of each eggplant and trim skin away from 2 sides, then slice all the eggplant into thin long pieces, cutting top to bottom into ¼-inch planks. Salt the eggplant and drain 30 minutes to 1 hour on layered kitchen towels.
Heat oven to 425˚F with rack in center. Line a couple of large baking sheets with parchment paper.
Heat a small pot or pan with about ½ cup olive oil and a large clove of crushed garlic over medium heat and let garlic gently bubble a couple of minutes. Brush the eggplant lightly with oil on both sides as you add to baking sheets. Roast eggplant 20 to 25 minutes, turning after about 12 to 15 minutes, until eggplant is golden brown and tender then remove from oven and cool to handle.
Meanwhile, heat a pot over medium to medium-high heat with EVOO, 2 turns of the pan, add onions and salt and soften, add black or red pepper flakes and remaining chopped or sliced garlic and stir, add ½ cup water and let it absorb. Add tomato paste, balsamic vinegar and combine a minute more, add tomatoes, passata and basil. Simmer 20 to 30 minutes.
Line a large casserole with a thin layer of the sauce, about ⅓, layer in half the eggplant, add more sauce to cover, then half the Parm and mozzarella, repeat. Cover with foil and bake for about 25 to 30 minutes at 375˚F then remove foil and bake another 10 to 15 minutes until bubbly and brown in middle of oven. Top with torn fresh basil.
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