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1
Soak the porcini mushrooms in the hot water until softened, about 20-30 minutes. Remove the mushrooms from the liquid, squeezing out the excess. Strain the soaking liquid through cheesecloth or a coffee filter to remove any grit, and set the strained liquid aside.
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2
Put the onion, carrot, and softened mushrooms in the bowl of a food processor and grind until they are chopped fine - you want the texture of a very chunky paste.
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3
Heat 1.5 tablespoons of the olive oil in a large, heavy bottomed pot over medium heat. Add the pancetta and cook until brown and crisp. Add the remaining oil and the onion/carrot/mushroom mixture, season with salt and cook until the vegetables are very soft.
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4
Add the chicken livers and cook just until they have lost their raw color, stirring occasionally. Clear a space in the bottom of the pot and add the tomato paste, allowing it to toast for a minute before stirring it through.
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5
Add the vermouth and wine and let them bubble up, and then add the reserved mushroom soaking liquid. Add the tomatoes, breaking them up with your fingers, then add the sage. Cover, reduce the heat, and simmer for 10 minutes, then uncover and continue cooking for 20 to 30 minutes more, stirring occasionally and tasting to adjust the salt as needed (you'll be adding cheese at the end, so don't overdo it).
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6
Stir in the grated Parmagiano Reggiano off the heat and toss with your hot cooked pasta. Serve with additional grated cheese and a sprinkling of fresh chopped parsley if you like.
