Preheat the oven to 400 degrees F or 200 degrees C.
Heat 1/8-inch (3mm) of oil in a big skillet over a medium heat.
When it is nearly smoking, add some slices of eggplant and cook, turning once, until they are browned on each side and cooked through, watching out for splatters.
This should take 5 minutes or so.
If you prefer, you can bake the eggplant.
Let the cooked eggplant slices drain on paper towels.
Put more oil in the skillet, heat it up, then add more eggplant slices until they are all cooked.
Meanwhile, in a small bowl, mix the ricotta with the egg, Half & Half, 1/4 cup of the parmesan, and a small pinch each of salt and pepper.
Arrange a layer of eggplant slices in each of 2 gratin dishes.
Sprinkle with parmesan and a little salt and pepper.
Spoon in 1/2 of the marinara or tomato sauce.
Add a second layer of eggplant, another pinch of salt and pepper, half the ricotta mixture, and then a tablespoon of grated parmesan on top.
Put both gratins on a baking sheet and bake for about 25 to 30 minutes or until the top is browned.
Sprinkle with fresh basil before serving if you like, for a nice splash of color.
