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for the pork: Remove pork from refrigerator one hour before starting to cook. Rinse and pat dry with a paper towel. Sprinkle all over with salt and pepper, fairly generously.
Preheat oven to 350° F.
Heat a 5 to 7 quart Dutch oven over medium to medium-high heat. Add oil. When oil is hot and starts to shimmer, add pork. Cook until nicely browned and then turn, repeating that same process until all sides are browned. Add broth, onion, oregano, cumin, and paprika. Turn heat up a bit to bring liquid to a boil. Then spoon some of the hot liquid over the top of the meat.
Place the cover on and transfer to preheated oven. Cook until meat is literally falling apart, about 4 hours, spooning liquid over meat once or twice during cooking.
for the sautéed sweet peppers: About half an hour before the meat is done, prepare the sweet peppers. Cut into long slices, about 1/4″ wide or so. Heat a wide skillet over medium heat - I like to use a large cast iron skillet. Add butter and olive oil. When the butter has melted, add the peppers, stirring to coat. Cook for about 10 minutes, letting them soften up. Then turn up the heat to medium-high. Let the peppers start to darken and char, stirring every couple minutes. Transfer peppers to serving dish.
to crisp the pork: When the pork has finished cooking, remove pot from the oven and place on stove top. Transfer meat to a platter. With a large spoon, push meat off the bone in chunks; discard bone and any large pieces of fat. Transfer pork back to French oven and turn on heat to medium-high. Cook, stirring with a large wooden spoon to break up pork, until liquid in French oven has almost entirely evaporated and meat begins to sizzle and caramelize, 5 to 10 minutes. The meat should pull into shreds very easily.