In a small bowl, soak the dried mushrooms in 1/4 cup of hot water for 30 minutes. Remove the mushrooms from the soaking liquid and coarsely chop. Strain and reserve the liquid.
Heat 2 tablespoons of the oil in a medium saucepan over medium heat. Add the carrots, celery, and shallots, and cook until softened, about 5 minutes. Add the garlic, porcini, rosemary, and cook for 2 minutes more. Add the tomato paste and cook, stirring, for 2 minutes. Add the wine and the reserved mushroom liquid and simmer until the liquid is reduced by about half. Add the tomatoes and a 1/2 teaspoon of the salt. Bring to a simmer, cover, and cook over medium-low heat for 20 minutes. In a large skillet, heat the remaining 1 tablespoon of oil and butter over high heat. When the foam begins to subside, add the mushrooms. Lower the heat to medium-high and cook, stirring occasionally, until golden, about 5 minutes. Add the remaining 1/2 teaspoon salt and season with pepper to taste. Add the mushrooms to the tomato sauce and simmer for 5 minutes. Serve as is, or with pasta, polenta, or toasted bread.
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