Preheat the oven to 350°. In a medium, ovenproof saucepan, bring 2 1/2 cups of water to a boil. Add 1 teaspoon salt, then slowly whisk in the cornmeal over moderate heat until smooth. Simmer, stirring, for 1 minute. Cover and bake in the oven for about 25 minutes, or until thickened; stir vigorously a few times as it bakes. Stir in the butter and Parmesan and season with salt and pepper.
Spoon the polenta into a lightly oiled 10-by-8-inch baking dish and smooth the top. cover with plastic and refrigerate until firm, at least 2 hours or overnight.
Preheat the oven to 450°. Cut the polenta into 8 rectangles. Divide the oil between 2 large, heavy-bottomed ovenproof skillets and heat until almost smoking. Season the chicken with salt and pepper and add to one of the skillets, skin side down. Arrange the polenta cakes in the second skillet. Cook both over moderately high heat until the chicken and polenta are golden brown on the bottom, about 8 minutes. Turn the chicken and polenta, transfer the skillets to the oven and bake for about 12 minutes, or until the chicken is just cooked through and the polenta is golden brown.
Transfer the chicken and polenta to a platter and keep warm. Set the skillet that the chicken cooked in over low heat and add the shallots. Cook, stirring, until softened, about 3 minutes. Add the tomatoes and cook, stirring, for 2 more minutes. Add the basil and season the sauce with salt and pepper.
If desired, cut the polenta cakes in half diagonally. Spoon the tomato sauce onto 4 plates, set the chicken and polenta cakes on top and serve at once. Leftover cakes can be refrigerated overnight and rewarmed in a 350° oven.
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