Preheat the oven to 425 degrees F (220 degrees C).
For the roasted pig: Mash together the salt, peppercorns, rosemary, garlic and thyme with a mortar and pestle or in a food processor. Slowly add the olive oil to make a paste.
Score the skin of the pig and rub with the herb mixture, making sure to get inside the skin. Place the pieces of meat in a large roasting pan. Add the shallots, garlic heads and 2 cups (500ml) water to the pan. Cover with foil. Roast for about 2 1/2 hours. Remove the foil during the last 30 minutes of cooking and baste often with the pan drippings until the skin is crispy and golden brown.
For the rapini: Saute the rapini in butter and season with salt and pepper. Keep warm.
Serve the roasted pig with a ladle of Mushroom Stew and the rapini.
Place the dried mushrooms in a bowl and cover with water. Let soak for about 15 minutes.
Heat some oil in a cast-iron casserole over medium heat. Add the shallots and saute until softened, about 5 minutes. Add the button mushrooms and cook until browned, about 5 minutes. Add the garlic, and then the ketchup, and continue cooking for a few minutes.
Strain the rehydrated mushrooms and reserve the soaking liquid. Give the mushrooms a quick chop and add to the pan. Cook for 5 minutes. Add the reserved soaking liquid and season with salt and pepper. Cook until the sauce has thickened, about 20 minutes.
To serve, sprinkle with parsley and garnish with pecorino.