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1. Grease the slow cooker with cooking spray.
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2. Place the eggplant and zucchini in a large colander and toss well with 3 tablespoons of the salt; let sit for about 45 minutes. Rinse well to remove the salt, then dry well, gently squeezing out excess water with a kitchen towel. Add to the greased slow cooker. Heat 2 tablespoons of the butter in a 10-inch, heavy saute pan over medium-high heat. When warm, add the onions, bell peppers, and garlic and saute until the vegetables are softened, about 8 minutes. Transfer to the slow cooker. Reduce the heat to medium and add the remaining 1 tablespoon of butter. As soon as it melts, add the flour and tomato paste, and cook until the mixture is thickened and the flour disappears, about 1 minute. Increase the heat to medium high and add the tomatoes with their juices, thyme, and pepper. Cook, crushing the tomatoes a bit with a wooden spoon, until thickened and smooth, about 6 minutes. Mix with the vegetables in the slow cooker. Cover the slow cooker and cook on low until the vegetables are tender, 4 to 5 hours. Uncover and remove from the heat; immediately stir in the cheese and basil.
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While the ratatouille is cooking, add the stock and salt to a medium, heavy saucepan and bring to a boil over medium-high heat. Once boiling, gradually whisk in the cornmeal. Reduce the heat to low, and cook, stirring every 3 minutes, until creamy and thickened, 30 to 35 minutes. Remove from the heat and stir in the butter, cheese, and pepper.
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Ladle polenta into individual bowls, spoon ratatouille on top, and serve immediately.
