In a medium bowl, combine the cabbage and 1 teaspoon salt; using your fingers, rub the salt into the cabbage. In another medium bowl, stir together the pork, garlic, ginger, 1 tablespoon soy sauce and 2 tablespoons water. Set the cabbage and pork aside.

Bring a large pot of water to a boil. Add the noodles and cook, stirring occasionally, until tender. Drain in a colander and rinse under cold water until completely cooled, then drain again. Toss with 2 teaspoons neutral oil; set aside.

In the same pot over high, heat the remaining 1 tablespoon neutral oil until shimmering. Add the pork mixture and cook, breaking it into small pieces, until starting to brown, 5 to 6 minutes. Stir in the miso until incorporated, then stir in the remaining 4 tablespoons soy sauce, the sherry and the scallion whites. Add the broth and 2 cups water. Bring to a boil, then reduce to medium-low and simmer, uncovered and stirring occasionally, for 10 minutes.

Squeeze the liquid from the cabbage back into the bowl; discard the liquid. Add the cabbage to the pot and cook, uncovered and stirring occasionally, until tender, about 7 minutes. Add the noodles and cook, stirring, until heated through, about 1 minute.

Off heat, stir in the vinegar, then divide the soup among 4 serving bowls. Sprinkle with the scallion greens and a generous drizzle of chili oil. Serve with additional soy sauce and vinegar.