Preheat oven to 375 degrees F. In a medium saucepan, combine the rice, 2 cups water, bay leaves, 1 tablespoon of the oil, and ¼ teaspoon of the salt. Bring to a simmer, cover, and cook until water is absorbed, about 7 minutes. The rice will still be al dente. Spread the rice onto a sheet pan to cool.
Layer the eggplant in a large colander in the sink, and sprinkle liberally with kosher salt.
Set a flat plate large enough to cover most of the eggplant slices, weighted with cans, to help press the excess liquid and bitterness from the eggplant. After about 20 minutes, rinse and drain the eggplant and pat dry.
Heat the remaining 5 tablespoons olive oil in a Dutch oven over medium heat. Add the onion and cook until softened, about 5 minutes. Add the tomatoes; slosh out the can or bowl used for crushing with 1 cup of water, and add that as well. Season with the remaining teaspoon salt, and bring to a boil. Nestle the eggplant slices in the sauce, and simmer until tender, about 15 minutes. Stir in the basil, and remove from heat.
In a medium bowl, toss together the two cheeses. Ladle about a third of the tomato sauce into a 9 x 13-inch baking dish. Fish out and layer a third of the eggplant in an overlapping pattern. Spread over that half the rice, then another third of the eggplant in overlapping slices, and a third of the sauce. Sprinkle with half the cheese. Top with the final layer of rice, then eggplant and sauce, and the last of the cheese. Cover the dish with foil, taking care not to touch the cheese, and bake until bubbly around the edges, about 25 minutes. Uncover, and bake until browned and crusty on top, another 25 to 30 minutes.