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1
For the pulled pork, heat the olive oil in a Dutch oven and brown the pork on two sides over medium heat, working in batches as necessary to avoid crowding the pan. As the pork is browned, remove it to a plate.
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2
Once all the pork is browned, add the onions, garlic, and ginger to the pan and cook, stirring occasionally to pull up all the browned bits from the bottom of the pan, until they begin to soften, about 5-8 minutes. Add in the star anise and mushrooms and stir for another minute more.
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3
Nestle the pieces of pork back into the pan. Add the soy sauce, wine, vinegar, water/stock, and sesame oil, and stir to distribute. Add in additional water or chicken stock as necessary to bring the liquid level to about halfway up the pieces of pork (it's fine if there's a bit more than that). Bring the liquid to a boil, then reduce heat to a gentle-to-moderate simmer and cover the pot.
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4
Braise the pork until very tender, about 2 hours.
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5
While the pork is braising, make the kale slaw. In a large bowl, combine the miso, lemon juice, vinegar, olive and sesame oils, and pepper to make a thick dressing. Toss in the kale, apple, and carrots, and mix continuously for a couple of minutes to coat the kale very well. Make it at least an hour or two before serving. Store in the refrigerator.
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6
When the pork is fork-tender and falling apart, remove it from the pan and onto a large plate. Remove any unwanted hunks of fat still remaining, and shred the meat with two forks (or your fingers if the meat is cool enough). Skim the fat from the pan and discard. Also discard the star anise pod. Reduce the sauce over medium-low heat to about 1/2 cup (if there is more than that).
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7
To assemble sandwiches, place a heaping mound of shredded pork on the bottom half of each bun. Drizzle a tablepoon or two of reduced sauce onto the meat. Top with a generous mound of the kale slaw. Top with the other half of the bun, and serve.
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8
If you like an extra-spicy, vinegary kick, slice 1-3 hot peppers (jalapeno, serrano, habanero, Thai, or a combination of) and set them in a cup of rice wine vinegar before starting the pork. Spoon the vinegar over the meat (as much or as little as you'd like) before adding the kale slaw on top.
