Heat the oven to 325°F with a rack in the lower-middle position.

In a large Dutch oven, combine the pork, ½ cup of gochujang, 2 tablespoons of miso, the cilantro stems, the hoisin, ginger and 1 cup water; stir to combine. Bring to a simmer over medium-high, then cover and place in the oven. Cook until a skewer inserted into the meat meets no resistance, about 3 hours.

Meanwhile, in a nonstick 12-inch skillet over medium-high, heat the oil until shimmering. Add the onions and ¼ teaspoon salt, then reduce to medium. Cook, stirring occasionally, until the onions are golden brown, about 15 minutes. Stir in the remaining 4 tablespoons miso and cook, stirring frequently, until the miso begins to brown, about 5 minutes. Transfer to a plate and let cool, then cover and refrigerate until ready to use.

Using a slotted spoon, transfer the pork to a large bowl. When cool enough to handle, shred into bite-size pieces, discarding any fat; set aside.

Remove and discard the ginger chunks from the cooking liquid. Tilt the pot to pool the liquid to one side and use a wide spoon to skim off and discard as much fat as possible from the surface. Bring to a simmer over medium-high and cook, stirring occasionally, until reduced by about half and a spatula drawn through the sauce leaves a trail, 5 to 7 minutes.

Whisk in the remaining 4 tablespoons gochujang. Stir in the pork and onions. Reduce to medium and cook, stirring frequently, until heated through, 5 to 10 minutes. Off heat, stir in the vinegar, then taste and season with pepper. Serve with cilantro leaves, pickled carrots and pickled jalapeños.