Coat an 11x7-Inch baking dish with 1 teaspoon of oil. Transfer the hot polenta to the prepared baking dish, spreading evenly to ¾ inch thick. Cover and refrigerate until cold and firm, about 2 hours.
Preheat the oven to 250 degrees F. Cut the polenta into 2x1-inch pieces. In a large, heavy skillet, heat the remaining ½ cup of oil over a medium-high flame. 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, and serve, passing the marinara sauce alongside.
When I make polenta I like to use a heavy-bottomed saucepan because it cooks the polenta evenly and helps develop rich, creamy, sweet corn flavor.