In a small bowl, stir together the tomatoes and ¼ teaspoon salt; set aside. Set a box grater in a large bowl or pie plate. Using the grater's large holes, grate the corn down to the cobs; reserve the cobs.

In a large pot, bring 2½ quarts water to a boil. Add the corn cobs and 1½ teaspoons salt, reduce to medium and cook, covered, for 10 minutes. Using tongs, remove and discard the cobs, then remove the pot from the heat.

In a 12-inch nonstick skillet over medium, melt 2 tablespoons of butter. Add the grated corn, shallots, chili and ½ teaspoon salt. Cook, stirring, until the shallots have softened, about 5 minutes. Stir in 1½ cups of the cooking water.

Cook over medium-low, uncovered and stirring occasionally, until slightly thickened (a spatula should leave a brief trail when drawn through the mixture), 10 to 15 minutes.

Meanwhile, return the remaining corn-infused water to a boil. Add the pasta and cook, stirring occasionally, until al dente. Reserve 1 cup of the cooking water, then drain the pasta.

Add the pasta to the skillet and cook over medium, stirring constantly, until the pasta is coated and the sauce is creamy, about 2 minutes; if needed, add the reserved cooking water 2 tablespoons at a time to reach proper consistency.

Off heat, add the remaining 2 tablespoons butter, the tomatoes with their juices and the basil, then toss until the butter has melted. Taste and season with salt and pepper.