Peel the eggplants and dice them. Toss the cubes of eggplant in a small amount of salt and put in a colander sitting over a bowl. Set the eggplant cubes aside to drain while you prepare the rest of this dish.

Prepare the custard by whisking the eggs and cream together. Stir ina few tablespoons of cheese and 1 T. of the chopped basil. Set custard aside.

Heat oven to 375 degrees F and oil an 8 by 10 gratin dish. (Alternatively, you can use 1 cup ramekins.)

Return to the eggplant and blot the moisture off the surface of the eggplant with a kitchen towel. Put 2 T. oil in a large nonstick skillet and turn on the heat to medium high. When the oil is hot, add the eggplant and cook it for 12-15 minutes until the cubes turn soft and are golden brown in spots. Stir this mixture while it is cooking to monitor it and keep it from burning. Scrape this mixture into a bowl and set aside.

Put 2 T. oil into the same pan that you cooked the eggplant in, return the heat to medium heat and add the chopped onion. Cook the onion, stirring occasionally, until it is softened and begins to take on a bit of color. This will take about 5 minutes. Add the minced garlic to the pan and cook until the garlic becomes fragrant. Add the tomatoes and the cooked eggplant to the pan and season with salt and pepper. Cook this mixture for about 5 minutes. Taste. Add salt if necessary. Transfer this mixture into your prepared gratin dish.

Pour the custard over the vegetable mixture and sprinkle with the remaining cheese. You can add a bit more cheese here.

Bake about 30 minutes. At the end of baking you can run the gratin under the broiler to get it a little more brown. Be careful, though, the browing occurs very quickly under the broiler.

Remove from the oven and cool for a few minutes before serving. Garnish with chopped parsley or cilantro.