For the soup: Season the chicken all over with the salt. Tie up the ginger in a piece of cheesecloth (or put in a tea bag or tea ball). Put the chicken and ginger in a large pot with the soy sauce, scallions, carrots and 8 cups water and bring to a boil over high heat. Reduce to a simmer and simmer about 1 hour, skimming the foam off of the top when needed.
For the matzo balls: In a large bowl, combine the matzo meal and 1 teaspoon salt. Stir in the oil, egg yolks, scallions and 1/4 cup of the soup. In a separate bowl, beat the egg whites until fluffy but not stiff. Fold the whites into the yolk mixture. Cover and refrigerate for 30 minutes.
Transfer the chicken to a cutting board and let cool slightly, 10 to 15 minutes. When cool enough to handle, use a fork or your hands to pull the meat from the bones; reserve the meat. Return the soup to a very low simmer while you make the matzo balls.
Bring a pot of salted water to a boil. Roll tablespoon-size balls of the matzo mixture. Boil in the salted water until fluffy, 30 to 35 minutes.
Return the meat to the soup and simmer until warmed through. When ready to serve, stir the black vinegar into the soup. Serve the matzo balls in the soup with a drizzle of sesame oil and a plop of sambal oelek.