Coat a large pan or Dutch oven with oil and place over medium heat. Add the chicken livers and sear until golden brown on all sides, about 7 minutes. Remove from the pan and set aside. Add the onions, carrots, celery and fennel to the pan and season generously with salt. Cook, stirring frequently, 3 minutes, then add the garlic and cook until the vegetables have softened, about 7 minutes. Transfer to a blender and pulse (see Cook's Note). Return to the pan and cook until well browned and the water has evaporated, 15 to 20 minutes.
Roughly chop the chicken livers and add to pot. Add the beef, sausage and beef heart to the pan and sprinkle with salt. Cook until the beef is browned, 15 to 20 minutes. Add the tomato paste and cook until it browns, 4 to 5 minutes. Add the red wine and let it reduce by half. Add enough water to cover the meat by about an inch. Add the thyme and bay and stir to combine. Bring to a boil, then reduce to a simmer and cook, stirring occasionally and adding more water if necessary if it evaporates, until the meat is tender, 3 1/2 to 4 hours.
During the last 30 minutes of cooking, bring a large pot of water to a boil and season generously with salt; the water should taste salty like the ocean. When the water reaches a rolling boil, add the pasta and cook for 1 minute less than the package directions.
Meanwhile, remove half of the ragu from the pan and reserve. Remove the bay leaves and thyme bundle.
Drain the pasta and reserve some of the cooking water. Add the pasta to the pan with half of the ragu and the reserved cooking water and continue cooking over medium heat until the water evaporates, for another minute or so. Mix in some Parmigiano-Reggiano. Serve with additional sauce.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.