Slice eggplants into ¼-inch thick planks and arrange in single layer on 1 or 2 baking sheet(s).

Sprinkle both sides with salt and let rest for 20 to 30 minutes to draw out excess moisture. Gently press the eggplants with clean dish towel to squeeze out moisture.

Meanwhile, preheat grill to medium heat. Clean and oil grates.

Brush one side of eggplants with 1/4 cup olive oil and place on grill. Reduce grill to medium-low.

Brush other side of eggplant with olive oil and cook, turning occasionally, until lightly charred and tender, 10 to 15 minutes total.

Meanwhile, in small saucepan, heat remaining olive oil over low heat. Add garlic and dried basil and cook for 30 seconds, then add the crushed tomatoes and stir.

Bring to a simmer and cook, stirring regularly, until slightly reduced and flavors are developed, 10 to 15 minutes, reducing heat as necessary to avoid burning. Season to taste with salt.

Transfer grilled eggplant to serving dish and cover with tomato sauce. Garnish with fresh basil and Parmesan. Serve immediately or at room temperature.