Heat the oil in a large deep skillet over medium high. Add the onion and cook until translucent - a few minutes. Then add the garlic and stir until fragrant - about 30 seconds. Add the eggplant and cook for 10 minutes or until tender. Adjust olive oil if more is needed - but don't overdue it. Stir occasionally. Season with salt and pepper to taste.
Next add the tomato, a little more salt and pepper, and toss for a few more minutes until warmed and slightly wilted. Sprinkle with your cheese and cover pan until it's melted. Serve with fresh basil.