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Grease the slow cooker with cooking spray.
Place the eggplant and zucchini in a large colander and toss well with 3 table-spoons 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 sauté pan over medium-high heat. When warm, add the onions, bell peppers, and garlic and sauté 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 (10 grinds). 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.
Make the polenta While the ratatouille is cooking, add the stock and salt (1 1/2 teaspoons) 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(4 tablespoons), cheese, and pepper (10 grinds).
To serve Ladle polenta into individual bowls, spoon ratatouille on top, and serve immediately.