Preheat a grill. Brush the eggplant, squash, mushrooms, and asparagus with the oil. Place the vegetables on the grill and cook over medium-high heat, turning the squash, mushrooms, and asparagus over after 4 minutes, and the eggplant after 5 or 6 minutes. Cook all the vegetables 4 minutes longer.
Remove the vegetables from the grill. Slice each mushroom into 8 strips. In a large pot over high heat, combine the water and salt and bring to a boil.
Whisk in the polenta or cornmeal in a slow, steady stream. Reduce the heat to low and cook, stirring, for 20 to 30 minutes, or until the polenta reaches a thick but spreadable consistency.
Preheat the oven to 375ºF. To assemble the casserole, put ½ cup of the marinara sauce on the bottom of a 13" x 9" baking dish. Add half of the polenta. Layer each of the grilled vegetables atop one another. Evenly spread the remaining polenta on top of the vegetables. Smother the top with the remaining marinara sauce. Sprinkle with the cheese and basil.
Cover with foil and bake for 30 minutes, or until the sauce starts to bubble. Remove the foil and bake for 10 minutes longer, or until the cheese browns. Let the casserole cool for 5 minutes before cutting.