Heat a grill pan over medium heat. Slice the eggplants into 1/4-inch rounds. Toss them with 1/4 cup olive oil and some salt and pepper. Grill them until marks have formed and they are soft, about 2 to 3 minutes per side. Place them onto a paper towel lined platter. Scatter over 2 cloves of sliced garlic, drizzle with 2 tablespoons of olive oil and the lemon juice, put on a few torn basil leaves, and season it all with salt and pepper. Set aside while you make the sauce.
Coat a large saute pan with olive oil and place over medium heat. Cook the onions until they are soft, about 5 minutes. Add the chile, garlic, olives and a few hand-torn basil leaves. Carefully add the tomatoes and a pinch of sugar; cook and stir until the liquid is cooked down and the sauce is thick, about 15 minutes. Add the capers, taste, and adjust the seasoning with salt and pepper. Lower the heat, cover, and keep warm.
Fill a large pot with water and bring it to a boil. Add a couple of tablespoons of salt and the spaghetti. Stir to make sure the pasta doesn't stick. Cover and return to a boil. Remove cover and cook for about 6 to 8 minutes, or until the spaghetti is al dente. Drain and add the spaghetti to the tomato sauce. Add the cheese and mix well. Pile onto a platter and top it with the eggplant slices. Garnish with more basil leaves.