Make the ratatouille: In a Dutch oven or heavy 4-quart pot with a lid, heat the oil over medium heat. Add the onion, stirring occasionally, until translucent, 4 to 5 minutes. Stir in the garlic; cook 1 minute more. Stir in the eggplant and zucchini, 1-1/2 teaspoons salt, and ¼ teaspoon pepper. Add 1/4 cup water, cover, and simmer, stirring occasionally, until the vegetables begin to soften, 5 to 6 minutes. Stir in the bell pepper pieces; simmer, covered, until they’ve softened, 5 minutes more. Stir in the tomatoes and the dried herbs; bring to a boil. Reduce heat to medium-low. Partially cover; simmer, stirring often, until vegetables are tender, 12 to 15 minutes.
Make the polenta: In a small saucepan, combine the chicken broth, salt, and 2 cups water. Bring to a boil over medium-high heat. Slowly add the polenta, stirring constantly with a wooden spoon. Lower the heat to medium-low and continue cooking, stirring often, until thickened and creamy, about 5 minutes. Add the butter and continue stirring until it’s incorporated, 1 minute more.