Put 3 tablespoons oil in a large, deep saucepan or casserole over medium-high heat. When hot, add onion, carrots, and zucchini. Cook, stirring, until onion softens and vegetables begin to caramelize, 10 minutes or so. Add corn and beans; sprinkle with salt and pepper. Cook, stirring, for a minute or 2, then add 6 cups water and tomato; bring to a boil, then adjust heat so mixture bubbles gently. Cook, stirring every now and then, until vegetables are fairly soft and tomatoes have broken up, about 15 minutes. Add 1/2 cup basil and adjust heat once again so mixture simmers. Cook until all vegetables are very tender, 5 to 15 minutes longer. Taste and adjust seasoning, drizzle with remaining olive oil, and serve, passing basil and cheese at table.