Chop up the reserved rabbit bones. Put the bones and butter in an ovenproof saucepan and roast until browned. Transfer the pan to the stovetop and set over medium heat. Add the shallots, garlic, sage, and rosemary and when they begin to turn golden, add the wine and cook until it evaporates. Add the water and simmer until reduced to one-third of its initial volume.
To make the spezzatino:
Preheat the oven to 350 degrees F.
Cut the reserved rabbit meat into 1-inch cubes and season with salt and pepper. Heat the butter in a roasting pan set over medium heat. Add the rabbit, and let it color on all sides. Add the garlic, shallot, rosemary, and sage, and cook just to develop a little color. Add the wine, let it evaporate, and then add all the rabbit stock. Roast, uncovered, until the meat is tender, 20 to 30 minutes (add water to the pan if the juices begin to burn). Transfer the meat to a plate with a slotted spoon and remove and discard the herb sprigs. Add the cream to the pan gravy and reduce over medium heat until it has thickened but is still a little runny.
Spoon some of the rabbit in the middle of a plate. Top with Funghi di Bosco. Try and turn the bucatini around the rabbit to make a nest (it won't completely stay in place), sprinkle chopped parsley on top, and finish with a drizzle of olive oil.
To make the dough:
Put the flour in a mixing bowl, make a well in the center, and break the eggs and yolks in the middle. Add the olive oil to the well and beat with a fork to incorporate the olive oil with the eggs. Slowly begin to assimilate flour until you have a thick paste and can begin working with your fingers and assimilate the rest. Put the dough on a lightly floured countertop and knead for a few minutes until the dough is elastic. Cover with plastic wrap and let the dough rest in the refrigerator for 30 minutes.
Bring a large pot filled with salted water to a boil. Drop the pasta in the water and cook until the bucatini is al dente (try one to check), 2 to 4 minutes. Put the butter and sage in a saute pan set over medium heat and let the butter solids turn brown. Add the bucatini to the pan along with a little pasta cooking water (this stops the butter from browning) and keep moving the pasta gently in the pan until the water evaporates. Add the cheese. Serve with the sage leaves, if you like.
Funghi De Bosco:
Preheat the oven to 400 degrees F.
Arrange the mushrooms in a baking dish. Melt the butter with the sage and garlic in a small pan set over medium heat. When the butter solids have turned a nutty blond, pour the butter (discard the garlic and sage) over the mushrooms. Season with salt and pepper and roast, turning occasionally, until the mushrooms are tender but still slightly al dente, 15 to 20 minutes. Transfer them to a bowl, toss with the parsley, drizzle with olive oil, and keep warm.
Melt the fontina in a microwave oven on high heat, cooking for 1 minute intervals, until the cheese is runny. Pour into a pastry bag and immediately roll out the bucatini dough. If the cheese hardens in the meantime, you can reheat it in the microwave for a few seconds, but only if the bag has a plastic (not metal) tip.
Roll out the bucatini dough using a pasta machine, always dusting the pasta sheet with flour to prevent sticking, until the sheet is 1/32-inch thick (or slightly thinner, if you can).
Cut the pasta sheet crosswise into strips 2 1/2 to 3-inches wide (and about 4-inches long). Brush the strips with egg white and squeeze out a rope of fontina stuffing down the length of the strip, stopping 1/4-inch from each end. Roll the pasta strip over the stuffing until you have a tightly stuffed cylinder. The pasta should roll over itself twice. Pinch the ends closed and then cut off the pinched part to seal the bucatini. Make at least 24 bucatini.