Lightly oil an 11 by 7-inch baking dish. Transfer the hot polenta to the prepared baking dish, spreading evenly to 3/4-inch thick. Refrigerate until cold and firm, about 2 hours.
Preheat the oven to 250 degrees F.
Cut the polenta into 2 by 1-inch pieces. Heat the oil in a heavy large skillet over medium-high heat. Working in batches, fry the polenta pieces until golden brown on all sides, about 3 minutes per side. Using tongs, transfer the polenta pieces to paper towels and drain. Place the polenta pieces on a baking sheet and keep warm in the oven while cooking the remaining batches.
Transfer the polenta pieces to a serving platter. Sprinkle the polenta with the Parmesan cheese and salt. Serve, passing the marinara sauce alongside.
Bring 6 cups of water to a boil in a heavy large saucepan. Add 2 teaspoons of salt. Gradually whisk in the cornmeal. Reduce the heat to low and cook until the mixture thickens and the cornmeal is tender, stirring often, about 15 minutes. Turn off the heat. Add the butter, and stir until melted.