In a bowl, soak the mushrooms in warm water to cover until softened, about 15 minutes. Drain. Thinly slice the mushrooms and set aside.
Bring a medium-sized pot filled with water to a boil over high heat. Add the noodles and cook according to the package directions. Drain, rinse with warm water, and drain again. Place the noodles in a bowl and cover to keep warm.
To prepare the meatballs, put the meat, cornstarch, sesame oil, salt, and pepper in a food processor and process to a smooth paste. Scoop the meatball mixture out into a medium bowl. With wet hands, roll the mixture into walnut-sized balls, using a heaping tbsp. for each ball. Arrange the meatballs, without crowding, on a lightly greased plate.
To make the soup, in a medium pot, combine the broth and vinegar and bring to a boil over high heat. Add the meatballs and return to a boil. Reduce the heat to medium and simmer until the meatballs are cooked through, about 8 minutes. Stir in the mushrooms and pickled greens and simmer for 2 minutes more. Add the cooked noodles and cook, stirring, until the noodles are heated through. Season to taste with salt and pepper.
Ladle the soup into bowls. Garnish each bowl with some of the green onions, cilantro, and chill oil, and serve.