For the pork shoulder: One day before cooking, liberally salt the pork shoulder and refrigerate uncovered.
When ready to cook, preheat the oven to 300 degrees F.
In a large Dutch oven over medium heat, cook the pancetta and 6 sage leaves over medium heat until the pancetta renders its fat and gets crispy, about 5 minutes. Using a slotted spoon, transfer the sage and pancetta to a plate lined with paper towels and set aside. Add the pork shoulder to the pancetta fat and brown the pork on all sides, about 5 minutes per side. Set aside.
Add the garlic, carrots, rutabaga, turnips and onion and cook until browned, about 4 minutes. Add the tomato paste, miso paste, rosemary sprigs and sage sprigs and cook for 1 minute more. Add the beer and chicken stock and bring to a simmer. Return the pork to the pan and add the reserved pancetta and sage. Cover and braise in the oven until fork-tender, about 3 1/2 hours. Allow the pork to rest for 30 minutes in the liquid.
For the fried parsley: Put 1/2 inch of oil in a small saucepan and heat over medium heat until 350 degrees F. Add the parsley leaves and fry until crispy, about 1 minute. Remove with a slotted spoon to a paper towel-lined plate.
For the fried onions: Put 1/2 inch of oil in a small saucepan and heat over medium heat until 350 degrees F.
Mix together the flour and cayenne in a shallow bowl. Dredge the onions in the flour mixture, shaking off any excess. Fry in batches until browned, 3 to 5 minutes. Remove with a slotted spoon to a paper towel-lined plate.
For the fried capers: Pat the capers dry with a paper towel. Fill a small saucepan with 1/2 inch of oil. Add the capers to the cold oil and turn the heat to medium. Cook until the capers are fried and crispy, 3 to 5 minutes. Remove with a slotted spoon to a paper towel-lined plate.
Remove the pork from the liquid and transfer to a cutting board. Slice the pork into 1-inch-thick pieces and transfer to a serving platter.
Stir the vinegar into the braising liquid. Pour the sauce over the pork and top with the fried capers, fried onions, fried parsley, mint and cilantro.
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