PREHEAT THE OVEN to 200°F.

TO MAKE THE OVEN-ROASTED TOMATOES: Brush a large baking sheet with olive oil and lay the tomatoes on it in a single layer, cut side up. Sprinkle them with salt and pepper, then turn cut side down. Brush the tops of the tomatoes with olive oil and sprinkle with salt and pepper. Roast for 4 to 6 hours, regulating the heat so that it doesn’t get so hot that the tomatoes burn. Since the tomatoes must cook at a very low and constant temperature, it is best to use an oven thermometer. Prop the oven open by 1 inch with a dish towel if the temperature is difficult to regulate at such a low heat. The tomatoes are done when they are shrunken, dehydrated, and shriveled on top but still quite juicy underneath. Cool on the baking sheet and wrap with plastic wrap until needed.

TO MAKE THE SAUCE: Rinse and drain the Swiss chard. Remove the leaves from the ribs, trim and discard the woody bottoms, and slice the ribs 3/8 inch thick. Coarsely chop the leaves and set both aside separately.

IN A LARGE HEAVY SAUTÉ PAN, cook all the sausages over medium-low heat for about 15 minutes, stirring occasionally, or until no trace of pink remains. With a slotted spoon, remove the sausage to a bowl, then pour off and discard the fat from the pan. Wipe the pan with a paper towel. Over medium heat, combine the olive oil, onion, and carrot and sauté for 6 to 8 minutes, or until the onion is translucent. Return the sausage to the pan, add the sage and garlic, and stir for 2 minutes. Add half the wine and half the tomato paste and stir until most of the wine has evaporated. Cut the oven-roasted tomatoes in half lengthwise and then again crosswise. Add them to the pan. Add the remaining wine, remaining tomato paste, and the ribs of the Swiss chard. Cover the pan and cook, stirring occasionally, for 10 minutes. Remove from the heat, cover, and set aside while you cook the polenta.

SOAK THE PORCINI MUSHROOMS in 1 cup of very hot water for 30 minutes, squeeze dry and finely chop them. Strain the soaking water. Set aside both mushrooms and soaking water.

TO MAKE THE POLENTA: In a large heavy saucepan, heat the olive oil over medium heat. Add the onion and garlic and sauté, stirring occasionally, for 5 to 6 minutes, or until softened. Add the porcini to the pan and stir for 1 minute more. Add the chicken stock, the mushroom-soaking liquid, and salt and increase the heat to high. When the liquid is simmering, gradually sprinkle the polenta over in a very slow, thin stream, whisking constantly in the same direction until all the grains are incorporated and no lumps remain. Reduce the heat to very low. Switch to a wooden paddle and stir every 1 or 2 minutes for 25 to 30 minutes, or until the mixture pulls away from the sides of the pan and the grains of polenta have softened. Stir in the butter, pepper to taste, and Parmesan and remove from the heat.

RETURN THE SAUCE TO MEDIUM-HIGH HEAT, add the Swiss chard leaves and cover. After 2 minutes, remove the lid and stir to mix. Repeat twice more and, after about 6 minutes when the chard leaves are tender, remove from the heat. Mound some of the soft polenta onto each of 6 heated dinner plates, make a well in the center and spoon the sauce over.