In large nonstick saucepan or pot over medium heat, heat oil. Add onion and bell pepper. Cook until softened, 5 minutes. Stir in garlic. Cook 30 seconds. Add carrots. Cover and cook 2 minutes. Add zucchini, chili powder, and cumin. Cook 1 minute. Stir in tomatoes. Bring to a boil, lower heat and simmer, partially covered, 15 minutes, stirring occasionally.
Stir in black beans, kidney beans, pinto beans, and chickpeas. Heat through. Remove half of the bean mixture and refrigerate or freeze in airtight dish to use for another meal.
Preheat oven to 400°F. In medium saucepan, bring 4 cups of the water to a boil. In medium bowl, stir together remaining 1 cup water, the polenta, and salt. Gradually stir polenta mixture into boiling water. Gently simmer, stirring constantly, until the polenta is no longer gritty, 5 minutes.
Spread half of the polenta over bottom of 8 x 8 x 2-inch square baking dish. Spoon the remaining half of the chili mixture over polenta. Set aside 2 tablespoons of the cheese. Sprinkle remaining cheese over chili. Spread remaining polenta over top. Sprinkle with the 2 tablespoons cheese.
Bake until filling is bubbly and top is golden brown, about 20 minutes. Let stand 15 minutes. Cut into rectangles and lift out with a spatula.