Remove the thick skin from the pork, leaving a thin layer of fat on top of the meat. Season heavily with salt and place in the refrigerator until ready to use, as long as overnight.
Preheat the oven to 350°F Place a deep saucepan over medium-high heat and add 2 tablespoons olive oil. Gently cook the onion, celery and fennel until they begin to soften, about 10 minutes. Add the stock and thyme and bring to a simmer. Season to taste with salt and pepper.
Rinse pork to remove excess salt, dry with a paper towel and add to seasoned broth. Cover and place in the oven for 90 - 120 minutes, until the meat just begins to pull away from the bone.
Cool meat and broth on the stove top for 30 minutes. Remove the pork and gently pull the meat from the bone, then tear the chunks into bite-size shreds. Place these in a large bowl.
Strain the broth into a separate bowl and then pour enough of it over the meat to barely cover. (Use the rest for soup.) Cover and store in the refrigerator until ready to use.
When ready to make, put a large pot of salted water over high heat and bring to a boil. While the water is heating up, place a large pan over medium-high heat and add the pork and covering liquid. Bring to a boil, then lower heat to medium and cook until the liquid is reduced by half. Add the butter and stir to emulsify into a sauce.
Meanwhile, cook the pasta in salted water according to the directions on the package, 10 to 12 minutes. When it is finished, drain and add to the sauce along with a splash of pasta water. Simmer for 1 minute, then add the lemon juice, half of the cheese, a tablespoon of olive oil and the parsley. Stir to incorporate.