Slice the eggplants lengthwise into roughly 1/4-inch slices. Lay them out on a baking sheet and sprinkle both sides with fine sea salt. Let sit for 30-60 minutes until beads of moisture form, then pat dry with paper towels.
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 2 cans (14 oz each) chopped tomatoes, 1 tbsp tomato paste, and 1 tsp granulated 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 skillet over medium-high heat. Working in batches, fry the eggplant slices for 2-3 minutes per side until golden brown and tender, adding more oil as needed. Transfer to a plate lined with paper towels to drain.
Preheat the oven to 350°F (180°C).
Spread a thin layer of the tomato sauce over the bottom of an 8x12-inch baking dish. Arrange a single layer of fried eggplant slices, scatter pieces of mozzarella and a sprinkle of Parmesan over them. Repeat the layers—sauce, eggplant, mozzarella, Parmesan—finishing with sauce, mozzarella, and Parmesan on top.
Bake for 35-40 minutes, until the top is golden brown and bubbling.
Let the dish rest for 10-15 minutes before serving.