Heat the oven to 375°F with a rack in the lower-middle position. Set a box grater in a large bowl or a pie plate. Grate the corn kernels against the large holes of the grater down to the cobs. Stand each cob on its base in the bowl and, using the blunt side of a chef’s knife, scrape the cob top to bottom, allowing the “milk” to fall into the bowl; reserve the cobs.

In a small Dutch oven, combine the corn cobs, 6½ cups water and 1½ teaspoons salt; simmer over medium-high. Reduce to medium-low, cover and cook for 10 minutes. Using tongs, remove the cobs, allowing any liquid clinging to them to drain back into the pot; discard the cobs.

While whisking, slowly pour the polenta into the corn-infused broth. Bring to a gentle simmer over medium-high, whisking often, then place, uncovered, in the oven and cook for 1 hour.

Remove the pot from the oven. Whisk until smooth, then use a wooden spoon to scrape along the bottom and corners to loosen any stuck bits. Stir in the grated corn. Return, uncovered, to the oven. Cook until thick and creamy and the granules are tender, another 10 to 30 minutes (timing depends on the polenta or cornmeal). Meanwhile, in a medium bowl, stir together the tomato, chili, vinegar, oil and ¼ teaspoon each salt and pepper; set aside until ready to serve.

Remove the pot from the oven. Whisk in the butter until melted and fully incorporated. Let stand, uncovered, for 5 minutes. Taste and season with salt and pepper. Whisk in additional water, if needed, to thin. Serve topped with the tomato mixture, basil and a sprinkle of pepper.